Interesting point. Thinking it through for myself, it seems important that the money be sent to everyone, not just the unemployed. Otherwise you'd get a weird situation where taking a job might not increase the amount of money you make - or even decrease it, if the rule is that you only get money if you have literally zero income. Giving money to everyone makes it much easier to view jobs as a sort of sliding scale where you work harder and get more money.
Alternatively, you could set the minimum wage such that every job pays more than the base wage. But you'd still need a rule to deal with part-time work.
We do already have this problem with high marginal tax rates (counting the drop in available welfare as part of that 'marginal tax') at the border between unemployed/low-income/middle class. This article gives some good examples of where it happens currently: http://www.heritage.org/research/reports/2013/01/effective-m...
The other side of that same problem is that people who fall into lower levels of income have a very hard time proving they deserve access to the meager welfare programs that keep the permanently destitute from burning down society in a fit of lumpenproletarian rage.
Alternatively, you could set the minimum wage such that every job pays more than the base wage. But you'd still need a rule to deal with part-time work.