Retina, 18 bytes

.+
$*
(^1?|11\1)+$

Try it online! Shamelessly adapted from @MartinEnder's answer to Is this number triangular? but with the base conversion included at a cost of 6 bytes.

Note that Is this number triangular? wasn't for some inexplicable reason required to support zero as a triangular number, so part of the adaption was to add a ? to make the leading 1 optional, allowing the group to match the empty string, and therefore a zero input. However, having now matched the empty string, the + operator stops repeating, to avoid the infinite loop that would happen if it kept greedily matching the empty string (after all, ^1? would certainly keep matching). This means that it doesn't even try to match the other alternative in the group, thus avoiding the match of 2, 6, 12 etc. As @MartinEnder points out, a simpler way to avoid that while still matching the empty string is to anchor the match at the start while making the group optional for the same byte count: ^(^1|11\1)*$.