It didn't work quite as planned. The economy in the 80s did on average grow a little better than the 70s, but not much, and when you include grow thru 1992 not any better at all. As a consequence of lower tax rates, revenues lagged behind GDP; and because GDP growth was not super, there was no dramatic increase in revenues; to the contrary there was a relative decrease.
However, Reagan coupled his tax cuts with a major expansion of military spending.
When you cut back revenues and increase spending, you get deficits. The same was true with Reagan as it has been for Bush. Under Reagan's policies, the deficits tripled to record levels, and the national debt more than doubled; and had quadrupled to $4 trillion by the time Clinton got in and reversed course with a major tax increase in 1993.
So yes, it is completely fair to credit Reagan for a good portion of the nation's debt.