Majority: In a reversal from before 2001, most Americans do not support criminalizing private, consensual sexual relations between members of the same sex...and probably also do not support criminalizing private, consensual oral and anal sex for the heterosexual majority either.
With the quick succession of states legalizing civil marriage between same-sex couples, it is easy to forget that private, consensual sexual relations between same-sex couples -as well as certain sex acts between opposite-sex couples in some states- were criminalized up until a decade ago. Under various names such as crimes against nature laws or sodomy laws, such laws made it a crime for a gay or lesbian couple to be intimate in the privacy of their own home.
The American Bar Association's model penal code revisions in the 1960s began to urge state legislatures to remove victimless moral crimes including consensual sodomy. Keep in mind that such laws often banned oral and anal sex for heterosexuals too -and breaking these laws could lead to a felony conviction, listing on a sex offender registry, years in prison, and fines. While the laws were rarely used against heterosexual couples, they were frequently cited in custody cases involving a lesbian mother and the heterosexual father of her children from a former relationship. They were also cited as rationale for denying domestic partner benefits and allowing gay student organizations under the premise that such actions would support criminalized relationships.
My home state of Kentucky provides an interesting case study via Wikipedia:
At that time, Kentucky law criminalized consensual sexual relations between people of the same sex, even if conducted in private. Specifically, the law criminalized genital-oral (oral sex), genital-anal (anal sex), and anal-oral (rimming) sex -but only between partners of the same sex. Such sexual activities between mixed-sex (male-female) couples were legal. Such conduct was a misdemeanor punishable by up to 12 months in jail and a fine of up to $500. Solicitation of same was also a misdemeanor, punishable by up to 90 days in jail and a fine of up to $250. Historically, Kentucky's sodomy statutes had changed over time. The 1860 sodomy statute criminalized anal penetration by a penis and applied to both male-female couples and male-male couples. Because the law focused exclusively on penile-anal penetration, consensual sex between women was technically legal in Kentucky until 1974. In fact, in 1909 the Kentucky Supreme Court issued a ruling in Commonwealth v. Poindexter involving two African-American men arrested for consensual oral sex. In this decision the court upheld that the then current sodomy law did not criminalize oral sex but only anal sex.
In 1974 Kentucky revised its statutes as part of a penal code reform advocated by the American Law Institute. While the American Law Institute urged states to decriminalize consensual sodomy and other victimless crimes, the Kentucky legislature chose to decriminalize anal sex involving male-female couples but to broaden the new statute to criminalize anal-genital, oral-genital, and oral-anal sexual contact involving same-sex couples (both male-male and female-female couples). Thus, the 1974 revised statute decriminalized consensual anal sex for mixed-sex couples but expanded criminalization of sexual acts to include both male and female same-sex couples.
Morrison, Matthew (2001). "Currents in the Stream: The Evolving Legal Status of Gay and Lesbian Persons in Kentucky". Kentucky Law Journal 89 (4).
Jones, Jeffery (2001). Hidden Histories, Proud Communities: Multiple Narratives in the Queer Geographies of Lexington, Kentucky, 1930-1999. Lexington, KY: University of Kentucky (dissertation).
Illinois became the first state to decriminalize sodomy in 1968. Various states dropped their sodomy laws until the movement stalled with the advent of AIDS in the early 1980s. It was not until 1992 when Kentucky's Supreme Court overturned the state's same-sex only sodomy law on state constitutional grounds that decriminalization proceeded. Using similar arguments made in the Kentucky case, a lawsuit involving a gay couple arrested in one man's bedroom by Texas police serving him for traffic violations came before the US Supreme Court. In a close 5-4 decision in Lawrence v. Texas, the US Supreme Court overturned the last remaining sodomy laws in 2003.
Interestingly, conservative Republican Virginia Attorney General, Ken Cuccinelli is running for Governor of Virginia. He has argued Virginia's consensual sodomy law is still in force and applies to all Virginians:
But regardless of the US Supreme Court's ruling, does our Average American want anal -maybe even oral- sex between consenting adults criminalized?
Well, considering that one study looking at sexually transmitted infections finds that one-third of a sample of over 12,000 heterosexuals report having anal sex, and three-quarters say they have had oral sex, enforcement of such criminalization would be challenging to say the least.
But what about only criminalizing homosexual sodomy? It turns out that Gallup polls from 1977 to 2011 show a steady increase in Americans stating they believe gay or lesbian relations -sexual relations and relationships- should be legal. In 1977 Gallup found support at 43%. By 2011 this had risen to almost two-thirds of Americans (64%). Most Americans (56%) also now say they find homosexuality to be morally acceptable -an increase from 40% in 1977.
So, no, according to polling most Americans would not be in favor of criminalizing sexual relations between members of the same sex.