Wow, I had to do some digging. Almost every secondary source agrees that Constantine did indeed abolish crucifixion, but I had trouble finding any primary sources for this.

However, I managed to find two references, the first is in Aurelius Victor, De Caesaribus 41.4:

Denique Constantinus cunctos hostes honore ac fortunis manentibus texit recepitque, eo pius, ut etiam vetus teterrimumque supplicium patibulorum et cruribus suffringendis primus removerit.

There is a second reference in Sozomen's Ecclesiastical History, "He took away by law the crucifixion customary among the Romans, from the usage of the courts." EH 1.8

So, in short, the Roman Empire did away with the punishment of crucifixion largely due to the conversion of Constantine. However they did not abolish other forms of death penalty, and there is some ongoing discussion in the secondary literature about gladiatorial games that was interesting.

edit: added the quotation from Aurelius.