I installed dimmer but the wall switch does not have a ground wire; is it safe?
Connecting the switch to the black wires is safe and correct if you used the proper wire nuts. You said you made sure the connected parts were covered with electrical tape. If that's all you used, it's not good enough. Go back and use wire nuts. 

Never cut off a ground wire. If you don't have a ground in the box just tuck the wire in there for the future. There is a chance you've got a ground there if metal conduit or armored cable was used and screwing in the switch would ground it but not the way it should be. The switch is safe to operate as is.