I'm going to expand this a bit later. For now, take a look at this article on cuts to home care. There are a few problems with this strategy. First of all, you will save money in the short run as people find ways of making do. Without the security provided by regular services however, in the longer run you're going to see people moving into expensive nursing homes and institutions. Some of this will save states money because it will shift the financial burden to the feds, this is simply cost shifting that is definitely a poor outcome for anyone involved but the states.
More generally this is a problem with balanced budget rules that work year to year rather than focusing on longer periods. Saving money today will increase costs in the long term but taking this into account isn't baked into the rules. If nothing else, this downturn should be a call for local, state, and the federal government to take a hard look at overall government budgets and to consider moving towards a more sensible long term budget process. Even writing balanced budget requirements to be on a 4 or 5 year rolling basis would be a big step forward.