I have never heard of a problem with direct deposit in terms of compromise of information. I have heard about folks entering their information incorrectly (leading to chaos) or the MSP picking up that information incorrectly (leading to you not seeing your payment until you squawk).
Certainly my preferred method of payment is through Paypal because that gives no one access to your bank account. I also like that I get notice that the money has hit the Paypal account and I can then transfer it on to my bank or spend it or whatever as it is cleared cash.
My second best is the direct deposit and some companies never notify you when money is sent or which jobs it was for. But that is usually easy enough to figure out and from the date on my bank account of receipt I can mark the jobs off as paid on my shop sheet.
By far the least interesting is payment by check because some companies have too much information on their check which might alert a teller to your mystery shopper status. I generally deposit especially those checks through the night depository. But checks can bounce and cost you money and there can be other snafus. I just noted tonight that a deposit I made has had $27 deducted from it for no discernable reason (there were no $27 checks) so I am having the bank revisit the deposit to see what the problem was. (I always make a photocopy of the checks I am depositing for my records and the written amount and the numeric amounts match and the sum of the checks equals the amount on the deposit slip, so I am mystified.) Checks also get "lost in the mail" and when I first got started my checks were apparently being sent to another shopper with a similar name by one company. Eventually it all gets worked out but checks have wasted more of my time than any other form of payment.