I use cyberduck on my Mac. It’s been working for my FTP needs for several years, now. When I got it, it was nag-ware – free, but from time to time we will ask you to pay. At some point, I decided that I used it enough that I ought to pay for it. It might have a memory leak if I leave it open for too long, but that’t a long time. And maybe that’s been fixed, now that I think of it.
I first got it when I was working on a project with “ducklings” in the name, so I was very happy to get it. 
On the Mac I (still) use NetFinder. I also have a copy of the even-more-venerable Fetch.
I’m only an occasional 2nd-platformish type user of Windows but the FTP app I used that I liked the most was called CoffeeCup FTP. I have the vague impression that they gooped it up with adware in versions subsequent to the one I downloaded as shareware, though.
Coriolanus,
I actually use WebDrive where I work and I agree that it is a cool tool. I like how I can connect to all of the sites I edit with a different drive letter, and can also access my OneDrive account. There’s a free trial here: Download WebDrive WebDAV Client, Cloud Client, FTP Client
Since everyone else is mentioning using the command line, I guess I’ll mention that I just use Explorer directly in most cases. Back when I actually used it often enough, I had shortcuts set up with the requisite information.
The only times I couldn’t use it, it was because there was a maximum number of connections, and Explorer makes a new one each time, so I had to use the command line for that.