I think that if you check Princess' brochures you will find that Princess' official policy does not permit you to bring alcohol onboard with you. However, what the official policy is, and how strictly it is enforced may be two totally different things. I suspect that unless you are trying to bring cases of beer or alcohol onboard with you, there should be no problem. They will not, however, allow you to buy bottles of liquor onboard and bring it back to your stateroom. Royal Caribbean does sell bottles of liquor on board (at prices that are competitive with what you will find in the ports) and allows you to bring those bottles back to your cabin.