If you show proof of employment and are honest and upfront many landlords might be willing to rent to you, but if that does not work... offer to pay for either of the following options:
You can purchase a Surety Bond which varies by state. You pay a fee to a bond company who basically guarantees that the property will be returned in move-in condition. Many larger apartment buildings use this bond here in Hoboken, NJ. The landlord gets a guarantee and does not have to bother with deposit and accounting rules for a deposit.
The other option is deposit insurance and renters insurance. Both offer the tenant and landlord protection for payment of rent and other security deposit type issues. By agreeing to show proof of these insurances pre-paid for the lease term would be a big help in getting an apartment.
Here is an article I found that talks about both issues: http://www.moving-and-more.com/articles/Renters-Offered-Security-Deposit-Options.html
Best of Luck.
Hudson Place Realty
132 Washington Street
Hoboken, NJ 07030