This Times Op-Ed argues that the US should join Australia, and other countries that make voting mandatory. Would this be constitutional? Well jury-service and the draft, other mandates that are closely tied to civic duty (assuming that argument is correct) have been upheld. This likely would be constitutional as well.
Rick Hasen has more in this article:
This article examines the plausibility and implications of a norm-based explanation for voting. Part I reviews rational choice models for voting, contrasts the rational choice models with the social norms hypothesis and examines the empirical evidence regarding the existence of a norm of voting. As Part I demonstrates, rational choice explanations have offered only a tautological explanation of why people vote: People vote when the psychic benefits of voting exceed its costs. Unlike the rational choice explanation, a norm-based explanation of voting can explain plausibly why some people vote, as well as explain aggregate changes in voter turnout over time. Although the norms hypothesis is plausible, evidence supporting the hypothesis is sketchy and may be consistent with alternative explanations for voting. The analysis in Part I illustrates a general proposition that norm-based explanations are about as easy to conjure up as they are difficult to prove.
Assuming that a norm of voting has served to overcome collective action problems for only certain groups in the United States, and assuming that the norm has eroded over time even among these groups because of a decrease in social connectedness, arguably the state should take on the role of social sanctioner of last resort through a compulsory voting law. Part II of the is Article examines the substitutability of state- and societal-based mechanisms for social control in the voting context. In particular, this Part considers whether compulsory voting laws could serve as a good substitute for a norm of voting. Part II demonstrates that state and societal-based methods of control are not always substitutable. Enactment of a compulsory voting law in the United States, even if desirable as a method of overcoming collective action problems, and even if proven effective as a means of increasing turnout in other states, is unlikely to occur because of a widely held libertarian belief against government interference in the decision to vote.
Gotta think this through more.