Cliff, for clarity, do you mean the North Yorkshire Moors, or are you using the term thinking it includes/is the Dales? (It happens, they are moors in North Yorkshire after all).
The Dales and the North Yorks Moors are distinct National Park areas separated by a fairly significant expanse of the Vale of York (the river basin of the Yorkshire Ouse and its tributaries). It's quite feasible to visit both from one base. However, where you might stay or fish could be significantly impacted by your answer. For my money, as someone that lived and worked at sites on the top of the Yorlshire Wolds and NY Moors for over fifteen years, as you travel east of the Vale of York, the more limited the coarse fishing becomes.
The type of fishing you would like to partake in could also make a huge difference to your choices. Unfortunately for visitors, day ticket fishing on both the Swale and Tees is rather limited. I know far less of the Ure, Derwent and Rye, but they too don't seem overly blessed by day ticket access.
Fortunately, there are many nice little market towns to stay in. Some are far more touristy than others. Until you nail down the area you really want to be, it's difficult to recommend anything. I love the area around Pickering and if you are wanting to visit the NYMoors rather than the Dales, it would be a great base. However, I wouldn't recommend it if fishing is a priority.
