I'm not even a woodworker really, but ... No. It's indoors, it's not even in the bathroom which is basically the only place you have to worry about humidity.

If you really want a cheap preservative, you can use mineral oil. I've restored the handles of most of my garden tools with just sanding and mineral oil, and they've stayed quite sound even being outside year round. I even did a raised garden bed like that three years ago and it doesn't show a spot of decay, even the parts covered by soil. And it gets watered three times a day so it's very wet.