The chance that the next pope will be gay, is at least 5 %.
My reasons for believing this are simple (some would even say simplistic). First of all, at least 5 % of all people are gay.* Second, there is no reason to believe that gays become catholic priests any less often than others. On the contrary, it seems reasonable that young men seeing there's no chance of living happily ever after with a spouse might find it easier to choose to become priests than heterosexual young men.** Third, there is no reason to believe that gay priests would do any worse in the Catholic church than other priests.
Therefore, as at least 5 % of the cardinals may very well be gay, there is at least a 5 % chance that the next pope will be gay.
How will having yet another gay pope influence the politics of the Catholic church? One might argue that having hidden their sexuality all their lives, the gay cardinals may have developed a strong homophobia. That does not make sense, however, because it's unreasonable to argue that the gay cardinals have hidden their sexuality any more than the straight cardinals have. Their sexuality has just been "disregarded" - even by themselves. Therefore, it may be the case that a gay pope may actually not be homophobic, so that the Catholic church's position on gays may be a little less vitriolic than it was under the last pope.
Another question is of course: if the next pope is gay, will we ever know? The answer to that is probably "no". The sexuality of the pope will not become an issue - just as with the previous pope (I have never seen any comments on whether he was gay or straight). But as with other gay men, the new pope's sexuality may certainly be an influence on the way he sees the world.
* This is a matter of faith, as research gives so many different answers - depending on the methods and on the exact definition of gay. However, 5 % seems to be a reasonable compromise...
** When today's cardinals were young, the choice of being "out and proud" gays were just not there.