jmc

We may multiply in base $3$ just as in base $10:$ anytime we get a number greater than $3,$ we record the remainder when the number is divided by $3$ and carry the quotient. $$\begin{array}{r}2012\\ \times 201\\ 2012\\ 11101\\ 1112112\end{array}$$ Therefore, the product is $\boxed{1112112_3}.$