It would seem nearly unbelievable that the President would let it go so far that he actually went to prison. Although it would undoubtedly be unpopular politically, if a sitting President were actually convicted of a crime he could pardon himself as soon as the guilty verdict was announced.
It's hard to imagine that it would ever get to this point though. You'd need a president who was so wildly popular that it was essentially impossible for congress to impeach him. It's hard to imagine the level of popularity that would require though.
Let's consider a concrete case: Richard Nixon. Before the Watergate scandal broke, Nixon was wildly popular. In the 1972 election, he got 520 electoral votes, to 17 for George McGovern. The popular vote wasn't quite so one-sided, but still won the popular vote by the largest margin in US history. Pre-Watergate, Nixon had to look about as close to unimpeachable as any president in history.
Shortly after the 1972 election, however, the Watergate scandal broke. Here's what happened to Nixon's approval ratings:
Although he resigned before he could be impeached, by the time he resigned his approval ratings had dropped to the point that there was no longer any real question that impeachment proceedings would not only have proceeded, but would have undoubtedly resulted in his being removed from office.
To make a long story short, almost regardless of how popular a president started out, when enough evidence became public to uphold a conviction, it's almost inevitable that his approval would have dropped to the point that he'd be impeached and removed from office first. As such, there is essentially no possibility of a sitting president being convicted of a crime.