Chances are you'll be travelling to Aysgarth Falls by car. If you do then the best car park is the one in which the information centre and cafe are located. It costs a couple of pounds, but is well worth it - and you cna also go for a wander with the car park as a base, as it's unlikely that you'll spend a couple of hours gazing at the falls themselves!
Well yes, public transport does exist. Just make sure you plan your day and don't miss the last bus...oh, and be prepared to squeeze in with the others in a Royal Mail postbus :))) Most buses between Leyburn and Hawes stop here it's just that there are so few of them. Check the website below for exact times as they vary depending on season. Those buses are operated by Dales & District for which you can get a "Freedom Ticket" to use for a whole day's travel in the Dales but you need to pay extra if you use one of the Royal Mail vans which sometimes operate the same routes as they have to deliver mail to villages and might as well transport some people. Just note that the Royal Mail vans are only for up to around nine people and when they are full, they are full. Therefore, if you rely on one of those, try catching it from one of its end stops. Usually, there is no problem getting on but you never know, it might be market day in Hawes...
As for the stops, this picture shows the Monument stop in Aysgarth but there is also a stop at the turn off for the waterfalls a mile east of the village. All buses stop at the Monument but not all stop at the waterfalls unless you beg the driver nicely. Also note that the stops in the area are usually only signed on one side of the road but the buses of course drive (and stop)in both directions!
Aysgarth is on the A684 between Leyburn & Hawes. The turn-off for the falls is clearly signposted and is just by the Palmer Flatts Hotel. The A684 runs the length of Wenselydale and most of the villages here are very picturesque and have a nice pub or two.