The Wii remote is supported, but just like with the original Wii Virtual Console, you're going to have to plug in a Wii Classic Controller into it in order to use it. Sideways Wiimote and Wiimote/Nunchuck combinations aren't allowed for Wii U N64 downloads, even if it would work just fine for some games.
If you want a recommendation, I've been enjoying revisiting Paper Mario, Ocarina of Time, and Super Mario 64 on my Wii U with this 3rd party Classic Controller equivalent that I bought early this year. Also works well with Wii mode Virtual Console downloads of titles like Wave Race 64, which is yet to appear on the Wii U proper (Something that a Wii U Pro Controller can't do).
This officially licensed controller appeared last Fall from well liked 3rd party controller manufacturer, Hori. It's a good fit for N64 games, can easily be found new, and will only run $20-$25 (Half that of a Wii U Pro Controller).
If you buy an official Nintendo Classic Controller Pro these days, you run the risk of ending up with a cheap counterfeit. The real deal is out of print, leaving you without the option of buying it new from a source that is assured of being safe.
Cheap knock-offs are even happening with Wii U Pro Controllers these days (Essentially the Wii U successor to the Wii's Classic Controller Pro). So if you buy a Wii U Pro Controller, make sure to buy it at a reputable retailer that isn't going to be peddling counterfeits like you might end up with off eBay, buying used at GameStop, or from a 3rd party Amazon seller.
Best off buying it new from a chain like Wal-Mart, GameStop, etc.