If anything, EDTA should reduce background degredation of the RNA.  When the RNA degrades on its own, the 2' hydroxyl is activated by divalent metal/alkaline pH to cause strand cleavage.  EDTA should chelate the divalent metals and therefore reduce activation of the 2'hydroxyl so you get less degredation.

You could however get degredation from RNAses - make sure you don't have any of those!  Do you see evidence of extensive degredation?  You can add RNase inhibitor, but if you have a lot of RNAses it can be overwhelmed (RNase inhibitor is a protein).  What are you using to transcribe your RNA?  If you are using T7 polymerase I really like the NEB high yield transcription kit (not the regular T7 pol).  If your template is really clean, you should get great yields.

EDTA will stop polymerases which are Mg dependent. Some nucleases are also Mg dependent so those will be inhibited by EDTA. However many RNases are not Mg dependent so those RNases would still have activity in the presence of EDTA. 

Given that RNA samples can be stored long-term in TE buffer, then no EDTA shouldn't cause degradation of RNA in the sample. So as previously mentioned, if you are getting RNA degradation in your samples, it may be due to metal-independant RNases being present