How do you open a bottle of prosecco that's sealed with a bottle cap, assuming you don't want to make a loud and tacky "pop" sound?
You're pretty much stuck with the "pop," unfortunately, although some bottles will be louder than others depending on the carbonation pressure.
By "bottle cap" you're referring to the closures also known in the beer and soft-drink industries as crown caps. These have begun to appear on bottles of sparkling wines, notably a few brands of affordable Italian prosecco, such as Mionetto Il Prosecco. They've also been adopted by a few Canadian wineries for higher-priced bubblies, including Flat Rock Cellars and Featherstone in Ontario and Haywire in British Columbia. We'll be seeing more of these caps because they seal in freshness better than cork and do away with the risk of mouldy taint that continues to plague a small percentage of wines sealed with natural tree bark.
When it comes to traditional mushroom-shaped corks, the elegant and safe removal method is to take things slow and easy – and to never lose your grip on the cork as it's pulled from the bottle. The sound should be more akin to a hiss or "pfft" than an explosion. Judging by the wording of your question, I'm guessing you're familiar with this low-key and suave technique.
However, when it comes to crown caps, you must resort to a bottle opener rather than the gradual twist of your wrist. It's tough to do this slowly enough to produce a hiss (though you can try); in most cases you're faced with a loud pop or bang that will send pets and children running for cover. I've been startled at the decibel level when opening such wines myself, though not so much by proseccos, which tend to contain less pressure than most finer sparkling wines. Your best strategies are to chill that bottle very well (ice-cold, preferably), which tames the effervescence, and to hand out ear plugs to your guests and let the dogs and cats out first.