The 1960s were a time of change in the local gay scene in New Orleans. Perhaps the biggest catalyst for the changes was the murder of a gay man (Fernando Rios) in 1958. Despite ample evidence against them and dozens of witnesses, three college students were acquitted. As a result, the community that had previously been almost exclusively underground and apolitical began to organize. The first steps of organization weren't political, however, they were in the form of Mardi Gras krewes. These krewes gave the gay community an avenue to network and build community in ways they'd never been able to before due to the risks associated with frequenting the few gay bars that existed in the city.

One of the traditions of Mardi Gras krewes are costume parties/balls and beginning in 1962, the first krewe moved its celebration from a member's home to a local school cafeteria, marking the first true-to-tradition costume ball they were able to have. In the next few years, several more gay krewes formed, widening the network throughout the city and, by extension, putting them more in the public eye. This was, to quote Monk, both a blessing and a curse.

From You Make Me Feel: A Study of the Gay Rights Movement in New Orleans:

In 1961, the newly formed Krewe of Petronius was the first gay krewe to be legally registered as a Mardi Gras krewe and obtained a state charter to hold their ball. In 1962, the Krewe of Petronius also became the first gay krewe to obtain a police detail to prevent harassment. However, not all of the balls went smoothly. While the Krewe of Petronius enjoyed their celebration free of harassment from outsiders; the Krewe of Yuga’s ball was subjected to a police raid and resulted in a number of men being arrested and placed in jail.

These differing experiences show that although the gay community in New Orleans was growing more socially aware, they were still subject to discrimination by those in authority and members of the heterosexual community.

Which is where we get to your question's answer. Unfortunately, just as in other parts of the US, there were anti-sodomy laws in Louisiana that were used to "name and shame" men who were arrested for homosexual activity (or the suspicion thereof). From You Make Me Feel again:

Throughout the 1960s and 1970s, the New Orleans Police Department’s Vice Squad would often send undercover officers into gay bars and clubs in order to entrap gay men by arranging sexual encounters with them. ... Undercover officers would enter the gay bars and clubs and solicit these men; if they agreed to the sexual encounter they were jailed for attempting to commit sodomy, a sentence that carried a maximum of five years in the state penitentiary.

While most of what went on was your run of the mill raids and general harassment, perhaps the most bizarre thing to come of the increased scrutiny on the queer community was that in 1967 a New Orleans businessman and civic leader who was openly gay (Clay Shaw) was arrested and charged with conspiracy in the assassination of JFK. (No, I'm not kidding.) In 1969 he was found not guilty by a jury of his peers because OF COURSE HE WASN'T GUILTY, but that didn't stop it from having an impact on his life.

So, the best answer to your question is that as long as homosexual men kept their contact with other men in the privacy of their own homes and out of the public eye, they were likely to stay off the Vice Squad's radar and not be bothered. The flip side of that was that frequenting gay bars or "hangouts" or gatherings with a krewe they were a part of - even if there was nothing more than talk and maybe dancing going on - significantly increased the likelihood of being caught in a police raid or an undercover sting. Those who were arrested had their name and the charge against them published in the newspaper, which was definitely not ideal, to say the least.

So, how did New Orleans compare to other progressive cities at the time?

New York in the Sixties was also home to an overzealous Vice Squad that conducted many raids throughout the decade. I'm sure you're aware that it was a police raid in 1969 that led to the Stonewall Riots and the subsequent emergence of the Gay Rights Movement into the spotlight. That was far from the first time that the Stonewall Inn had been targeted by a raid, nor was it the only bar/club/hangout that was raided with regularity. This document from NYC's LGBT Historic Sites Project discusses many places within the city that were subject to raids and entrapment stings, as well as the general queer history about those places.

Chicago was in pretty much the same place despite the fact that Illinois was the first state to repeal their anti-sodomy laws in 1961. This didn't stop police from raiding gay bars under other pretenses, however. For example, they might raid a gay bar under the "suspicion" of mob activity taking place there despite the fact that they had no reason to think that was the case. This could still lead to people being arrested and having their names publicly attached to a bar that everyone knew was frequented by gay men. Raids and harassment peaked around the time of the 1968 Democratic National Convention held in Chicago because Mayor Daley really wanted his city to look good. (Spoiler alert: he didn't do a great job of making the city look good.)

In San Francisco - generally known for being the progressive city as far as LGBT+ communities and their rights are concerned - you would find many of the same problems as the cities above in regard to police and other agencies going out of their way to cause misery in the early 1960s. That said, things changed there faster than they did elsewhere and by the end of the 60s, it was well on its way to becoming what we know it as today. This was largely thanks to a strong presence by the Daughters of Bilitis (DOB) and the Mattachine Society, which helped with organizing the queer community to fight for their place in the city. Del Martin and Phyllis Lyon, a longtime lesbian couple who were co-founders of DOB, led the way on a tremendous amount of the progress that was made. (They were also the first couple in San Francisco to be married when same sex marriage was legalized in the state in 2008.)

The gay establishments in Los Angeles were also subject to raids by the police, one of which led to a gay "riot" a year before Stonewall that consisted of gay men marching into a police station carrying bouquets of flowers and just staying and making a nuisance of themselves until the police finally released those who'd been arrested in the raid. Very different tactics, but effective! The Mattachine Society I mentioned above was founded in Los Angeles as was the first gay-produced magazine (ONE). As with the presence of Mattachine and DOB in San Francisco, their organization and networks gave strong support to those fighting for gay rights in LA, even though it wasn't until the early 1970s when the community really found its footing.

Something you didn't ask about, but that I think it's important to mention, is that while the harassment levels and tactics of police in various cities were fairly equal through the 60s, the public attitude toward homosexuality was very different. New York and San Francisco especially had populations more tolerant to members of the LGBT+ community in comparison to New Orleans. The ideas that drive proper expression of masculinity in the South have historically tended to be more rigid than in the North and the response to transgressions against those ideas more pronounced.

That didn't stop cities like New Orleans (and Atlanta!) from having thriving gay communities from the 1950s onward, but it did often make them less welcoming to those communities. This, in turn, could sometimes lead to the police/DA's office/etc. to be less willing to take crimes against queer folks as seriously as they should have. That's kind of the other side of the coin where the impact of police presence on the gay community is concerned.