That is really a very subjective question, but I usually bring about $50 cash per day for souvenirs, taxis, food and other incidentals while in port. For larger purchases I use my credit card. There is an ATM onboard, but there will also be ATM's available at all the ports. You can also get a cash advance posted to your shipboard account at the casino.
I bring as little cash as possible. More or less the only thing I use cash for is tipping (on the lines operating a traditional tipping policy). Otherwise, I use traveler's cheques or my credit card. Traveler's cheques in particular are a good alternative to cash IMHO.
These days many people do not travel with cash. However, personal spending does vary. Once you are on the ship you will have no need for cash as lines use a cash card that acts as your room key, ID, cash. You can pay for this with cash, credit card, some lines will take a personal check. Tipping can now be charged to your room account too. Prior to this change I usually put my tip money aside before leaving. I also keep lots of singles for cabs, shopping, etc.
About tipping with a credit card. For most of the people that you'll need (or want to tip) this is the only "salary" they receive. If you are pleased w/ their service maybe you will consider tipping with cash in lieu of credit card. Mainly because they will have to wait sometimes one-three weeks for the transaction to filter back to them. Just a suggestion.