It's hard to say general opinions, especially over the long course of "the Troubles". Much of the funding for Catholic paramilitaries (I know less about the funding of the Protestant Paramilitaries in Northern Ireland) was through the diaspora, especially from places like Boston and New York. One of the famous IRA funders was Boston gangster Whitey Bulger. At certain points, the diaspora, especially the American diaspora, was the primary funder of the IRA.

Indeed, in Boston today you can still see a few IRA murals like the famous political murals in Belfast and Derry. Here's, for example, an old mural in South Boston. Notice the words "Sinn Fein" "Noraid" and the Gaelic on the top and bottom (the bottom Gaelic I think is "Óglaigh na hÉireann" which means "warriors of Ireland"), the Celtic Cross, and the shield emblems of all four provinces of Ireland (which is to say, including Ulster, six counties of which currently make up Northern Ireland). Just a wonderful mix of Republican iconography in the diaspora. Notice also that it explicitly says NORAID.That's also not the only example of a Republican mural in Boston. For more on funding and similar things, see the old thread, How prevalent was American funding of the Irish Republican Army?.

But the above only deals with a specific subset of Americans, the very committed, mostly Irish Catholics, mostly living in ethnic enclaves. I've met Americans who would swear to me that 26+6=1. On the other hand, there were also sympathies with the Northern Irish Protestants. My mother told me that, in her schools in California in the 1960's, one could wear green or orange on St. Patrick's day. The different colors didn't hold any significance to her (her family was neither Catholic nor Protestant), but clearly at one time some Protestants "protested" by wearing orange. When my mother told me that years ago, I tried to figure out how common it was to wear orange on St. Patrick's day in America, but didn't get very far.

But none of that gets at the general sentiment of Americans. The best way to assess that is opinion polling. The only polls I can find are only from 15-20 years ago, as the conflict was ending, but I think they're still historically useful.

A July 1998 Gallup poll found the following (link to Google Books):

In the situation in Northern Ireland, are your sympathies more with the Irish Catholics, or more with the Irish Protestants?

• Irish Catholics 30%

• Irish Protestants 16%

• Both (volunteered) 8%

• Neither (volunteered) 22%

• No opinion 24%

Every group where you'd expect to find Irish immigrants, you have more people sympathetic with the Irish Catholics: Whites, Easterners, Democrats. There is unfortunately no break down by religion, but I suspect Catholics were by a large margin more sympathetic to the Catholics and Protestants more evenly split. There aren't huge household income effects, but college educated were more sympathetic with the Catholics. Conservatives are split pretty evenly between Catholics and Protestants, but Liberals and Moderates lean heavily pro-Catholic (34% to 13% and 33% to 14%, respectively). Protestant support was fairly stable through all age groups (15-18%), but young people were much more likely to have a pro-Catholic opinion and much less likely to support "neither" (40% of 18-29 year olds supported Irish Catholics, while only 18% of 65 and olders did).

The same survey also asked another question:

What would you, personally, prefer to see happen in Northern Ireland--for it to remain part of the United Kingdom, or for it to become united with the Republic of Ireland?

• Remain part of the United Kingdom 17%

• Unite with the Republic 50% (!)

• None; other (volunteered) 4%

• It doesn't matter, don't care (volunteered) 14%

• No opinion 15%.

Here we actually see slightly different patterns of support: there's a clear gradient for higher income people supporting joining the Republic, for example, and more equal support for doing the Republic across age groups (the strongest support is now among 50-64 year olds) and ideological groups (the differences between Conservatives, Moderates, and Liberals are not statistically significant).

This was seen at the time as a relatively important issue(this is shortly after the Good Friday Peace accords, brokered in large part by the Clinton Administration--I'm very curious if anyone can find polling from a decade or two earlier). Gallup asked in February 2001 and again in February 2002:

As far as you are concerned, should the development of a peaceful solution to … be a very important foreign policy goal of the United States, a somewhat important goal, not too important, or not an important goal at all?

In 2001 21% and 2002 24% of Americans said that the solutions to the Catholic-Protestant conflict in Northern Ireland was "very important" (in 2001, 43% and 2002 54% said that solving the Israeli-Palestinian Conflict was "very important" and in 2002, shortly after the mutual nuclear tests in 1998 and during a peak of tension, 47% said solving the India/Pakistan conflict was very important, to give you some comparisons).

So, at least at the end of the Troubles, a good proportion of Americans had well developed opinions on the issue, and while there were Americans sympathetic to both sides, Americans were generally more sympathetic to the Catholic/Republican side, likely because of the large Irish-American population sympathetic to that side. However, there were also a sizable number of people who didn't think the issue was particularly important.

I would like to add to the answer by u/yodatsracist through a slightly different approach, largely regarding the political history involved. I would also like to preface my answer with an assumption that, because the question passes the 1996 limit imposed (i.e. The Troubles, as they lasted until 1998) my answer may also delve into this historical territory as, if not, the answer cannot be "in depth and comprehensive" as this subreddit requires.

It is for the very reason(s) you outlined in the question as to why there is arguably no tangible general consensus amongst Americans regarding the violence in Northern Ireland from 1968 to 1998. From L.B.J onward, the American political establishment (with exception to Bill Clinton and George Mitchell, who I will cover later) maintained a fine balance between refraining from antagonising the Irish/Semi-Irish population of America and maintaining the often delicate relationship with Britain - the so called "special relationship". To side one way or the other would have negative consequences to the opposing side; this is the inherent paradox which faced American policy makers, the President(s) included, during "The Troubles".

This attitude was persistent up until 1968, if not for the reasons previously stated then for reasons regarding the special relationship as it stood. For example, LBJ was on notoriously poor terms with Harold Wilson, Prime Minister from 1964 to 1970, regarding his refusal to commit British troops to, what he believed to be, an unjust war in Vietnam. With this relationship already at a strain, pushing his influence into the issue of Ireland, an inherently domestic issue for Her Majesty's Government, may have broke the camels back. Perhaps another example of this lies with the Iron Lady herself, Margaret Thatcher, and her relationship with Ronald Reagan. Although their relationship was the opposite to that of Wilson and Johnson, her renowned Iron will prevented her accepting any outside influence on the issue of Ireland. Indeed, after her own attempted murder by the IRA, the night before the Conservative Party conference in Brighton (12 October 1984; 02:54), her approach was quite clearly one (even if not stated in public) lacking negotiation. Despite signing the Anglo-Irish Agreement in 1985 (A document that did very little to stop the violence associated with 'The Troubles' and more so dealt with delegation of power), her approach toward Ireland was so opposed to negotiation that she had Gerry Adams voice dubbed over by an actor, as to prevent 'her' population from hearing his voice. Both of these examples, at two ends of the 'special relationship spectrum', if you will, illustrate how an American influence upon the issue of Ireland may have made matters worse for the Anglo-American relationship.

However, another important reason for a lack of Anglo-American work regarding Ireland is much simpler - this wasn't a foreign venture nor was it a domestic economic problem (the things that most often brought the USA and Britain together). 'The Troubles' were an inherently domestic issue, one that originated surrounding ethnic and nationalistic concerns, chiefly regarding the English and the Irish - Ireland being a traditionally ethno-nationalist state where the nationality is in fact defined by the ethnicity of the country's inhabitants. To this end, despite the Irish presence within America, dating back to the 18th Century, the issue was entirely without American origin, nor did America, in any tangible sense, have a great deal to gain by getting involved, especially as they couldn't secure a sense of victory, as America often had with their superior military or economic power.

Despite these factors, in 1998 the situation changed. Tony Blair and Bill Clinton, after only 1 year of partnership, were reportedly on very good terms. As the Monica Lewinsky story emerged, so to did Tony Blair in support for Bill Clinton, so much so that weeks after the fact, Blair was in Washington standing next to the president, reassuring the press of the strength and integrity of his character. After this event, their relationship changed and, perhaps for the first time in recent history, the American president owed a great deal of favours to the British Prime Minister. One of the favours asked by Blair was for Clinton to exert his influence in the Good Friday agreement in September of 1998, both personally and through Senator George Mitchell. The agreement was successful in securing peace and is widely regarded as the end to 'The Troubles', despite occasional violence thereafter.

Now, all of this may seem to be off-topic political history regarding Britain, Ireland and the United States. However, those involved in what I've discussed, I.e. the respective leaders of both Britain and America, geared policy decisions regarding Ireland and in this sense decided (if that isn't too Big-Brother) how informed the vast majority of the public were on 'The Troubles'. To this extent, although Britons often had an opinion regarding Ireland's troubles as it was an ethnic, nationalistic, domestic, religious, military and highly politicised issue, Americans often lacked an opinion (and therefore a 'general sentiment') regarding 'The Troubles' as American politicians had dodged the issues for over 40 years (for the aforementioned reasons) and thus the public were not informed. Those Americans who had a sentiment regarding Ireland were probably Irish-Americans, who largely supported the nationalist IRA and American politicians who supported a peace process and tried not to side either way, as to maintain the aforementioned balance between their Irish population and the special relationship.

I hope that this provides some new information regarding your question along with a fresh perspective not just on what may have been the sentiment of Americans (or, as I've argued, a lack thereof) but why there was/wasn't one.

Sources:

Marr, Andrew - A History of Modern Britain

Marr, Andrew - New Britannia (Documentary, very good series generally, focuses upon American involvement in Good Friday talks and thereby American sentiment)

AQA History: The Making of Modern Britain 1951-2007

Hennessy, Peter - The Prime Minister

If this post gets removed but anybody actually wants this answer for keeping, just PM me and i'll send you a copy.