I have an old Acer Aspire 3680 and would like to do a frugal install of Tahrpup to the hard drive, I have booted into it from usb and really like it because it is small and runs entirely on RAM. My only concern is that Puppy distros run as root users and therefore seem less secure than other distros. I have been searching around the internet and have seen many sides of whether or not puppy can be considered safe, and all that's done is left me confused. So my question is: Is puppy linux safe, if not, is there a way to make it safe?

You've really answered the question already. If it's running as root then it is more exposed so don't do that.
A lot of embedded systems run this way. But those systems have very limited capabilities usually and so aren't exposed the way a desktop system would be.

But really your question is too broad.

If you asked "Is this car safe?" How would people likely respond?
"Well if you go 150 it is not safe, but around town it's fine."

What you do with it can be a big factor in "safety".

Only run it as read only, as a live session for banking or whatever. If you do a frugal install. Run as user spot or fido.