Let me rephrase your proposal. Every eligible voter gets a $10 stipend. In order to vote, you must pay $10.
This is a poll tax, and the US has a long history of this. To summarize, this approach to provide monetary penalties for voting is generally viewed unfavorably because it disproportionately impacts the poor (who the money would benefit more).
In the US, the 24th amendment forbids this practice.
The right of citizens of the United States to vote in any primary or other election for President or Vice President, for electors for President or Vice President, or for Senator or Representative in Congress, shall not be denied or abridged by the United States or any state by reason of failure to pay any poll tax or other tax.
The government paying (crediting) nonvoters is the equivalent of taxing at the polls, and would have the same results.