The down arrow button and the "OK" button on my remote are only working sporadically. All other buttons seem to be working correctly. The remote is attached to a non-smart TV that is over 10 yrs old. Is it time to purchase a new remote? Or is there a way to convince these two buttons to work again?
You could try taking the remote apart and cleaning the contacts etc, assuming you tried multiple sets of batteries and rebooting your 3600, etc.
However, you likely need a new remote: Either the $20 Voice Remote (RCAL7R) or $30 Voice Remote Pro (RCS01R) can be used, and can be obtained from Roku.com/Walmart/Best Buy/Amazon/etc.
https://www.roku.com/products/accessories/streaming-stick
However, considering the age of your model 3600 and the fact that you can get a new $30 Express 4K+ or $40 Streaming Stick+, I recommend just getting a new Roku: