You may be able to remove the mold from the old cover. I've had some limited success with a Lysol wash, followed by lemon juice and salt, on moldy tents and boat covers. I've also had some fairly good results by soaking them in a deck cleaner that was made for removing mold, brushing the heck out of it, pressure washing, then running them through a washing machine.
Mold will develop on any cover if you cover the ski wet. The key is to let the ski dry before you put on the cover. Same advice applies to boat covers. Problem is that this is not always practical. If you are trailering, leave the cover off on the way home. At the dock let the ski sit and dry before covering, or use a towel to dry it first.
To prevent mold an anti-mold spray on the inside of the cover will help, but may not completely prevent it (Moldex is one of the more popular products).