However, the woman is asking if she can pay the deposit in increments, instead of in full up front?

Be professional business woman.

I've come across this situation too many times: good tenant, single woman often mother of children, deadbeat baby daddy get's out of prison moves in and ruins everything.

(real case)But I'll take any day someone who filed bankruptcy for medical bills - people not going to die because they were not able to pay $100K for a surgery.That's why I meet every tenant and talk to them about everything: job, family, dogs, weather.....lolI want to get their character and life in general.I almost took application from a woman - she was nice and sweet and her income was sufficient with no evictions etc.

It opens up you and me to a lawsuit.I have 3 condos that I cannot sell because some idiot didn't put smoke alarms in his condo and a woman and her two kids died theren his property.

This woman was a stay at home mother. 

And just a few years ago, a woman was murdered showing a vacant house two doors down from my in-laws' place in Arkansas.

As a result her benefit will be cut about $400/month.Now the woman is really tough spot.

I know my area well and did ok with stocks as a young woman.

I absolutely agree, but here you have a situation where the woman is unresponsive to the landlord's calls, there's an active leak, and she's locked hereself into the bedroom when the ladlord came in.