My choice so I do not consider making appointments
with providers who ask for them.  I understand the need for it to be done from standpoint of the individual provider, and don't have problems with that - it is THEIR choice to do so.

nor have I been asked to.  Generally these are only required when you are scheduling 8+ hours of a ladies time, or ask a lady to travel with you.  This is simply because any last minute change of plans on your part, means an actual and/or potential loss of revenue for her.

In cases whree a deposit is appropriate, how best do you pay them - however she asks you to so.

your bank account to hers. Before doing any of that you need to insure that the lady is for real and has a good reputation for honesty and service.
Sometimes deposits can get you time with ladies that are great and will not visit unless they are sure of the business that they are targeting. Deposits can often be the next best thing to flying the lady in to see you. The important key is to make sure that the lady will not take the deposit and vanish, there are plenty of those types operating in this business so be careful.

I have paid a deposit one time. I paid it with a money order. Sent it Via UPS so I could track the shipment. I had no problems with this method.

I understand not wanting to send in one to an unknown, or to somebody who has not yet established herself with a positive reputation.  But does anybody seriously think that any lady who has a good reputation would jeopradize that for a few hundered or few thousand dollars?  I am honestly wanting to hear what any of your concerns may be because I honestly can't come up with a single reason why a deposit would be a negative thing.

In answer to the original poster - I'm sure the lady in question will let you know her preferred method or methods - and you can select the easiest and most discreet method for you form her list.