Political scientists argue that there are two factors that affect whether people register and vote. These are the costs of voting and the benefits gained from voting. If the federal government is to increase participation (which is not really what seems to be happening right now--there are more moves towards reducing it) it will need to either reduce the costs of voting or raise the benefits.
The most plausible things the government could do would be to reduce the costs of voting. These are not monetary costs but rather costs of time and effort. The government could, for example, do away with the need to register beforehand. Same-day registration would allow people to vote without having to go get registered at some previous date. It could also (as in my state, Washington) move to mail-only voting. This makes it so a person does not have to go to a specific place on a specific day in order to vote. If the government wants to keep in-person voting, they could make election day fall on a weekend or make it a holiday as some other countries do. All of these sorts of steps would reduce the cost of voting in terms of time and effort.
One related proposal that has been getting some attention lately is to make voting mandatory. This is unlikely to happen in the US, but it is an option.