I'd be very careful about trying to "determine their income is stable" period, because that's a subjective standard and can open you to legal problems. You need to stick to quantifiable, objective standards that are applied equally to all possible tenants and that don't unfairly favor one class status over another.
The answer about length of employment is problematic and in some places, might be illegal. In DC, for example, you can't choose someone over another simply because they've been at their job longer because that's a type of employment-based discrimination. What about actors who work for a different production company every few months? Writers who are self-employed? People who have independent funds and don't have to work? Students?
Even credit-checks can run into problems, though they're not a bad idea as one criteria among many. It shouldn't be your sole factor because young renters might have low credit scores because they have a shorter credit history. If you continually choose older renters with higher credit scores over younger renters you might be de facto discriminating based on age, which is a protected class if the only thing different between the scores is the length of time they've had access to credit. If you use a credit score, set it very low for outright rejection, and then have another, higher standard whereby you require the lease be co-signed with someone with a better score or charge a higher security deposit, if your state allows that.
What you can usually safely do is require that the tenant demonstrate a certain level of monthly income, say 3x the monthly rent. That's been the standard I've seen and I've rented six different apartments in three states over the last years. In many places, you can't favor one source of income over another (ie, a stable job vs. social security benefits vs. trust fund vs. savings vs. self-employment income).