Democrats actually controlled both Houses at the time and did much better in State houses and governorships back then.
Part of this was Democratic dominance in local, state, and Congressional elections in the South. Even as the South had become solidly Republican at the presidential level, this didn't really change until the "Republican Revolution" of 1994. Of course, a lot of these Southern Democrats were conservatives who were largely supportive of Reagan's agenda.
Still it's kind of remarkable that, given the magnitude of both of his victories, Reagan never had a Republican House. Indeed the Democrats controlled the House for 40 years straight starting with the 1954 election, even though the Republicans controlled the presidency for the majority of that time.
Reagan's coattails in 1980 were long enough for Republicans to capture the Senate, but Democrats took it back in the 1986 midterms.