No, they didn't. Though our evidence is a little unclear.

Christianity technically doesn't become the 'state religion' until Theodosius in the late 4th century. But it is earlier, under Constantine, that crucifixion begins to disappear.

Apparently he outlawed it. Howeer we don't have a direct legal basis for that. Instead, the church historian Sozomen mentions this in Ecclesiastical History I.8.13, "He took away by law the crucifixion customary among the Romans from the usage of the courts."

This appears to be supported by Aurelius Victor, De Caesribus 4:14, though this text is less explicit in saying that it is crucifixion that is banned.

It does appear that there was a general trend in the 4th and 5th centuries to change some form of punishments. For example, condemnation to death by fighting in the arena appears to have fallen out of favor too. So, as much as it probably should be linked with the rise of Christianity, it may also be seen as part of a broader move away from some forms of death penalty as 'barbarous.'