Is it safe to keep Bluetooth always enabled in my smartphone?

It will drain your battery

And, depending on the model, OS release, and your preferences/settings, might be open to unsupervised pairing / file system spelunking

That's what worries me. I want to keep a smartwatch paired with my phone all the time so I can get silent notifications.

If your phone is good, it will only pair when you tell it to

Some older Android, Windows Phones, possibly iOS (not 100% sure), and Blackberry had some exploits around Bluetooth

More-or-less any current phone (with current updates) should be safe from that.

Key being: it needs to be kept up-to-date