All I did was shoot a low level beaver with tranquilizer darts and feed it some citronals. It took me quite some time, but I got it. I do think it should be able to build beaver dams after taming. If you’re wondering how many narcotics I used, I used somewhere around 100 narcotics. I can’t remember how many I used. I think it was level four when I tamed it, and I was using citronals on Mobile ARK. I would recommend overpacking with narcotics and citronals because you don’t want to accidentally run out of supplies mid-tame. Hopefully this helps you get your very own beaver! :)
The method I used to tame was picking up the beaver with my crystal wyvern and dropping it in a taming pen. Pretty easy knockouts but their torpor drop rate is INSANITY and you can't leave then for more than 3 minutes. Feed them either mejoberries or crops (kibble is really expensive but if you want to do that, go for it).