In theory, 1000base-t was designed to work, even at a full 1Gbps, with cabling to basic Cat 5.
In practice, I have personally found that 1000base-t works perfectly, at a full 1Gbps, with basic Cat 5, with an error rate of zero over many many days.
If installing new cabling then obviously use 5E or better. But if you already have Cat 5, and you only need 1Gbps, it is highly unlikely that you will see any difference from using 5E,6, whatever.