Yes, mosquitoes do, indeed, become trapped on the silk of spider webs. How often do mosquitoes suffer this fate? We don't know the answer, but we do know some of the factors that relate to the frequency of this kind of event.
The webs of spiders are incredibly diverse in structure, and the form of the web depends upon the kind and size (age) of the spider. Many spiders weave webs with spaces between the strands that are just right to catch a large juicy insect, but allow smaller 'snack-sized' meals to fly right through. It takes energy and time for the spider to weave the web, and the more threads per unit area, the greater the expenditure. Would it be worth creating a densely-woven web just to catch small prey? If relying upon small insects in the diet were a good economical practice, then it is likely that the spiders would subscribe to that strategy.
Many mosquitoes are endowed with scales on the body and wings. Much like those on moths and butterflies, these scales are easily lost through abrasion.... and from contact with sticky substances, such as spider silk, tree sap and nectar. Many a mosquito that makes contact with a web will continue on its way, minus a little weight and ornamentation.
Note that not all spiders rely upon webs for ensnaring their prey. Certain kinds are active predators and will hunt and pounce upon their quarry. Jumping spiders are most adept with this strategy, and they are known to 'take down' mosquitoes with some regularity.
Oddly, many folks fear spiders and spider webs far more than they fear mosquitoes and the pathogens they transmit. Perhaps, it would behoove us to foster spiders in and around our homes. The nets they weave might provide as much or even more protection than do many of the commercially produced bed nets (at least the way those are improperly used in many homes).