There are other, more spiritual reasons why I accept modern prophets, but the following are some more objective, intellectual, historical reasons that work for me:
1) I have some limited experience with them that others haven't had, albeit in my case through my mother - who was one of Pres. McKay's secretaries many years ago. They really are amazing people.
2) I have a strong belief that THEY believe they are prophets, seers and revelators - and that means a lot to me, since it means they truly are striving to hear the voice of God in their lives and their callings.
3) I don't expect them to be perfect or infallible, so I don't hold them to that standard - but I think they "get it right" FAR more often than they get it wrong.
4) I believe adamantly that there isn't a "bad person" among them - that, if it is possible for there to be prophets on the earth, these are the kind of people they would be.
5) I believe, to a large degree, that prophets are as prophets are sustained and supported - that the expectations and hopes and dedication of the membership actually helps in a real way.
6) I can read the Bible and Book of Mormon -- and accept unmarried, Christian-killing, sexist-teachings accepting Paul -- and quick-tempered Peter -- and burn-up-the-competition Elijah -- and dysfunctional family father Lehi -- and perhaps bi-polar Nephi -- and (how can I describe him charitably . . . other than stupid for love or lust) Samson as prophets -- (as well as Muhammed and perhaps Jethro [not an Israelite from whom Moses received the Priesthood] and many others outside of Mormonism) -- so I certainly can accept our current First Presidency and Quorum of the Twelve Apostles.