Republicans own the national government and eventually may have to answer to the voters.

At the national level, it looks like the Democrats have no power. Republicans run all branches of government. About the only leverage the Democrats have is to use the filibuster rules in the Senate to shut down government.

Another strategy would be to avoid shutdown and let the Republicans rule till the voters get upset with things like rising health insurance premiums. Voters might toss out Republicans and / or put pressure on the ruling party to better fund the health insurance needs. Fearing voter backlash, some Republicans might break ranks and vote with the Democrats on various issues.

Lower courts can still check the power of the president, but that is subject to the Supreme Court in the long run which is basically run by the Republicans also.

State and local government, here in Washington State, is still functioning as normal. The private sector is still functioning and can put a check on absolute presidential power, such as in the case of media. Think Jimmy Kimmel.