1. Once you've been accepted by the landlord as a tenant you can explain your (justifiable) concern to your new landlord as ask to be allowed to contact his lender to determine the standing of his note. No, he's not obligated to give you the information, but if he's current, why wouldn't he?
2. You can visit the County Recorder's office and ask a clerk to help you determine if a Notice of Default has been recorded against the owner.
3. If the property is in Alameda County, you can visit thefollowing link and perform a search on the owner's name to see if a NoD has been filed against the owner: http://rechart1.acgov.org/search.asp?cabinet=opr
Be aware that the loan may not be in the owner's name. It may be in a business name, the name of a trust that may make it almost impossible to check. Your best bet is to discuss your concern with your new landlord.