They wouldn't need a secular justification. A religious one would have been just fine.
Let me explain.
It is not true that the First Amendment forbids laws "based" on religion. What the First Amendment forbids are laws "respecting an establishment of religion, or prohibiting the free exercise thereof." Lawyers split this up into the "Establishment Clause" and the "Free Exercise Clause." The Establishment Clause means that the government cannot, as a general matter, pass laws favoring one religion over another. The Free Exercise Clause means that the government cannot, as a general matter, pass laws restricting one's freedom to worship as one pleases.
It's hard to see how a law banning same-sex intimacy would violate the Free Exercise Clause (unless same-sex intimacy is part of your religion, in which case, send me a brochure.) Since the Free Exercise Clause isn't relevant, I won't really address it. The core of your question, I think, is more along the lines of: why didn't laws banning homosexuality violate the Establishment Clause?
The Supreme Court has, over the years, articulated some very complicated tests for determining whether a law violates the Establishment Clause. Since this isn't a law school class on religious liberty I won't really dig into the details of the test itself, but here it is:
In order to be valid under the Establishment Clause, a law:
Must have a significant secular (i.e., non-religious) purpose,
Must not have the primary effect of advancing or inhibiting religion, and
Must not foster excessive entanglement between government and religion.
Lemon v. Kurtzman, 402 U.S. 602 (1971). This is the so-called "Lemon" test. If it sounds vague and hard to apply, well, it is.
Notice what's not in the test, though. There is nothing in it that says legislators/Congress cannot consider religious beliefs when passing laws. The laws themselves cannot advance or inhibit religion, but legislators are absolutely allowed to let their religious beliefs to influence the laws they write. Legislators can invoke the teachings of the Catholic Church about the importance of forgiveness when passing laws that reduce prison time for drug dealers. They could invoke the Qur'an when passing laws limiting the amount of interest banks can charge. They can invoke Mormonism in writing laws banning the sale of alcohol on certain days. And they can invoke Leviticus to ban gay people from having sex with each other.
Said differently: just because a law has underlying religious motivations doesn't render it invalid. Indeed, religiously inspired laws are a huge part of American legal history. We've had laws about the Sabbath; laws restricting alcohol sales on Sunday; laws against abortion; laws forbidding bigamy; laws banning adultery; laws restricting contraceptives; laws giving you Christmas off; laws giving you Rosh Hashanah off (if you're in New York City); laws giving religious institutions tax exemptions; and many, many more. For the most part, the ability of legislatures to pass laws influenced by religious beliefs has been uncontroversial throughout American history. Even though the principle of religious liberty is enshrined in the Bill of Rights, America was a fairly religious country and saw no issue in laws that were influenced by religion.
There was some challenge to this principle in the 20th century, but those challenges generally failed. In the case of McGowan v. Maryland, 366 U.S. 420 (1961), the Supreme Court took up a challenge to a Maryland law forbidding the sale of most goods on Sunday (a "blue" law). The Court recognized that such laws were passed originally for religious purposes, but that over time they came to also have secular purposes (providing a day of rest, most prominently), and thus were permissible. Now, this decision came down during the most liberal era in Supreme Court history, the so-called "Warren Court" years (named after the Chief Justice at the time, Earl Warren). Even this very liberal court articulated a principle that basically said "if you can find a secular reason for your religious-looking law, you're good."
For a shockingly homophobic extension of this principle, I invite you to read Bowers v. Hardwick, 478 U.S. 186 (1986) a 1986 Supreme Court opinion that upheld anti-sodomy laws.
The majority opinion stated the following:
[The challengers to the law said there is no justification for the law] other than the presumed belief of a majority of the electorate in Georgia that homosexual sodomy is immoral and unacceptable. This is said to be an inadequate rationale to support the law. The law, however, is constantly based on notions of morality, and if all laws representing essentially moral choices are to be invalidated under the Due Process Clause, the courts will be very busy indeed. Even respondent makes no such claim, but insists that majority sentiments about the morality of homosexuality should be declared inadequate. We do not agree, and are unpersuaded that the sodomy laws of some 25 States should be invalidated on this basis.
In Justice Burger's infamous concurring opinion, he wrote:
As the Court notes . . . the proscriptions against sodomy have very "ancient roots." Decisions of individuals relating to homosexual conduct have been subject to state intervention throughout the history of Western civilization. Condemnation of those practices is firmly rooted in Judalo-Christian moral and ethical standards. Homosexual sodomy was a capital crime under Roman law. See Code Theod. 9.7.6; Code Just. 9.9.31. See also D. Bailey, Homosexuality and the Western Christian Tradition 70-81 (1975). During the English Reformation when powers of the ecclesiastical courts were transferred to the King's Courts, the first English statute criminalizing sodomy was passed. 25 Hen. VIII, ch. 6. Blackstone described "the infamous crime against nature" as an offense of "deeper malignity" than rape, a heinous act "the very mention of which is a disgrace to human nature," and "a crime not fit to be named." 4 W. Blackstone, Commentaries *215. The common law of England, including its prohibition of sodomy, became the received law of Georgia and the other Colonies. In 1816 the Georgia Legislature passed the statute at issue here, and that statute has been continuously in force in one form or another since that time. To hold that the act of homosexual sodomy is somehow protected as a fundamental right would be to cast aside millennia of moral teaching.
As an aside: one of Justice Burger's clerks at the time was a closeted gay man. (A clerk is a recently graduated lawyer who serves as an assistant writer/researcher/sounding board for a judge for a year or two--being a Supreme Court clerk is considered one of the legal profession's highest achievements). Imagine having reached the very top of your profession, working for one of the finest legal minds in America...and he makes you help him write an opinion calling you a disgrace to human nature and worse than a rapist.
So here we have the Court explicitly saying that religious beliefs can, indeed must, influence the law. And if that means that Judeo-Christian beliefs about gay sex being bad means making it illegal, so be it.
Now, you might argue--plausibly--that anti-sodomy laws violate the Establishment Clause because they fail the Lemon test: what is the significant secular purpose, and doesn't it advance religion, and doesn't it foster entanglement between government and religion? It's not a terrible argument! But as far as I know, this argument never really gained traction as a legal method to strike down anti-gay legislation. Frankly, gay rights advocates had stronger arguments to make, and make them they did.
American anti-sodomy laws were struck down in the great case of Lawrence v. Texas, 539 U.S. 558 (2003). (Shockingly late, I know...) if you read the case, you will find that nowhere does it mention the First Amendment, let alone the Religion Clauses. Anti-sodomy laws were struck down on the grounds of "substantive due process," essentially finding that sexual intimacy is a core liberty and that the government didn't have a good enough reason for criminalizing consensual same-sex sexual activity. (Con law lawyers who are reading this, I know this is something of a simplification, but I'm not trying to give a 1L lecture here). As far as I know, no one really made an Establishment Clause argument in Lawrence, simply because it wasn't as strong as the Due Process arguments (or the Equal Protection arguments, which the Court didn't take up for...well, complicated and probably homophobic reasons, but I won't get into that here).
Now, at least one serious scholar did argue that the laws forbidding same-sex marriage violated the Establishment Clause, but the Supreme Court ended up striking down these laws on a combination of Due Process and Equal Protection grounds in Obergefell v. Hodges, 576 U.S. 644 (2015). As far as I know, no serious Establishment Clause argument was made in Obergefell either-the Due Process arguments in Lawrence were easily extended to gay marriage, and the Equal Protection argument (that the state is treating same-sex couples differently than hetero couples, much like some states used to forbid interracial marriage) was also pretty strong.
EDIT: I deleted references to the Free Exercise clause and the tests for it because they weren't relevant, and because they get really complicated in a way that isn't helpful to answering the question.
EDIT 2: Snappier intro.
So, this is going to involve a little trip down American Constitutional history. Before we start, we'll need to understand the structure of the Constitution as it was set up in 1787. The U.S. Constitution sets up a federal system. You'll remember this as the concept of federalism, one of two ways the Constitution divides power, with the other being the separation of powers. We can think of federalism as a "vertical" separation of powers between the federal government and the states, as opposed to the "horizontal" separation of powers between the branches of the federal government (President, Congress, and Supreme Court). Of particular importance here is that the state governments make most criminal law, with federal criminal law typically requiring some jurisdictional "hook" like traveling across state lines or something similar.
Next, a brief word about "incorporation." As we'll see shortly, Prior to the Civil War the Supreme Court held that the Bill of Rights did not apply to the states. A state could set up a state church if it wanted (and several did so). It is not until the 14th Amendment is passed after the Civil War that applying the Bill of Rights against the states is even considered. Although there are some ups and downs in incorporation (see the Slaughterhouse Cases, ) the Supreme Court ultimately incorporates the majority of the Bill of Rights through the Due Process Clause of the 14th Amendment.
The First Amendment to the U.S. Constitution provides, in part, that "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof. . . ." As you can see from the text of the Amendment, it actually does not preclude the state governments from making an establishment of religion, and several states did so for years after ratification! The Supreme Court unanimously endorsed the interpretation that the Bill of Rights, including the First Amendment, did not apply to the states in 1833. (Barron v Baltimore, 32 U.S. 243 [1833].) So any sodomy laws would be perfectly legal, even if they were based upon a religious justification, up until the incorporation of the First Amendment's Establishment clause in Everson v Board of Education (330 U.S. 1) in 1947.
What about after 1947? Now that the Establishment Clause is incorporated against the states, surely sodomy laws can no longer be based upon religious justification, right? Well, as with any legal development, it's more complicated. The Supreme Court was faced with the constitutionality of a statute criminalizing consensual sodomy in 1986. The majority held the statute constitutional, but did not mention religious justifications for the law. Instead, the majority reasoned that only fundamental liberties "deeply rooted in this Nation's history and tradition" were protected from the states through the Due Process Clause of the 14th Amendment. (Bowers v Hardwick, 478 U.S. 186, 192 [1986].) A right to homosexual conduct was not among those fundamental liberties, as homosexual conduct was criminalized in the English Common Law and in all 13 states at the Founding. (Id., at 192 - 94.) The dissent took issue with this line of reasoning because the prohibitions found in the English Common Law and 13 states were founded upon religious justifications which would be precluded under the First Amendment. (Id., at 211-212.)
The dissent's view that prohibitions on homosexual conduct ran afoul of the First Amendment ultimately would not carry the day, as the Court held in Lawrence v Texas that sodomy laws were unconstitutional based upon a Due Process right to privacy. (Lawrence v Texas, 539 U.S. 558 [2003].)
This is something of a long-winded way of saying that, legally speaking, sodomy laws were perfectly constitutional until Lawrence v Texas in 2003 for varying reasons. This is not necessarily a political justification for sodomy laws, but it does show that the Establishment Clause was not (and technically still is not) a barrier to the passage of sodomy laws.