Well, the best thing for both of you would be to prevent any shedding problems in the first place. Offer a humid hide and keep it moist. Of course, don't stress if the gecko doesn't want anything to do with the hide, both of the leos that have lived with us for any extended amount of time won't have anything to do with it. For us, our ambient humidity is usually around 50% or higher year round; we live in a lake filled, snowy, climate and have a lot of problems with mold it is so humid! For our current leo, she loves her water bowl. We have a wide & shallow faux stone bowl, we have placed some plants leaning over the top in effort to hold in some humidity. We check her every shed and she's never left a piece of skin.
Also realize that if you want a tame gecko you should be thinking of interacting with it approximately once a week. (Give it a week or two when it first comes home just to itself, especially if it was shipped, just to let it settle in.) So a baby gecko sheds, let's say once every 2 weeks or maybe 3 as it gets older. It is not handling it too much if you check it after each shed. In fact, you really need to practice checking it after a shed so that it gets used to being inspected. Much like they do to get zoo animals comfortable with the vet before they have to be taken to a vet in an emergency.
Firstly, a Q-tip dipped in slightly warm water may be less stressful and can be brought into the enclosure, but obviously don't go chasing it around meanly. You could also consider replacing some of the paper towels in the bottom of the terrarium with soaked ones... our gecko tends to splash so much in her water bowl she does this herself, so she gets credit for the idea.
To be honest, cross that bridge when you get to it, because you may never have to worry. IMHO a well kept and healthy gecko can take care of their own shed 99% or more of the time. A moist Q-tip gently stroking the stuck part takes care of the other 1%. Hope this helps and don't worry. We'll be here if you run into problems!
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0.1 Albino Leo Gecko
0.1 Crested Gecko
1.0 Dear Boyfriend
Departed: Harvey and Spock