For a beard transplant, what can I apply to my skin (safe/won't affect hair growth) to combat dry crusty skin on recipient area?

You should not have crusting after a week. If you're still having crusting you need to see your doctor.

If the "crustiness" is your skin, stop pouring water on it!  That will actually strip the oils off your skin and make it more dry!  If it is your grafts, ask your surgeon.  Usually after 2 weeks, the grafts are pretty set, so some gentle scrubbing will take away the scabs, but check with your surgeon first.  If it's just dry skin, any gentle moisturizer will work.  Baby oil will also help in the short term, and help get rid of the scabs.

Following the procedure, one can expect to have small crusts formed on the grafted area. A post operative cleaning regiment should be given to the client to help dissolve the scabs. The cleaning should start the next day. These crusts will flake off by 7-10 days after the surgery. The client can usually return to normal activity and normal washing in about two weeks. The transplanted hair growing from the scalp may initially shed, however the roots will remain and new hair will start to grow in after three months.  Depending on  each hair intrinsic growth cycle , it may take up to 18 months for all the hair to grow in.