There really isn't much else you can do to specifically assess mRNA quality using surface level techniques such as bioanalyzer or denaturing PAGE. If you have a test gene that you know expresses under the condition you are studying (along with known pre-mRNA processing events), you can specifically look at its RNA products by Northern (to see mature and pre-mRNA products) or use qRT-PCR as a control to monitor RNA quality of your prep.
I do recommend running a denaturing PAGE gel of your RNA to assess quality of your RNA. You'll be able to determine if your prep contains any RNAses based on the banding pattern
The BioAnalyzer RNA 6000 Pico chip type can be run in mRNA mode. This is done by choosing 'eukaryote mRNA pico' as the chip type in the instrument software when setting up the run. When this is selected, the BioAnalyzer will report the mRNA quality by describing the percentage of rRNA contamination, instead of giving a RIN. (RIN is calculated by the ratio of 18S to 28S rRNA, and so when rRNA has been selectively degraded or removed, RIN is meaningless anyway.)
Percentage rRNA contamination is very useful when you want to determine how effective your mRNA isolation or rRNA depletion was, and whether it's comparable between samples. It doesn't tell you very much about whether your mRNA is degraded. The best way to do this on the bioanalyzer is by eye - you can find images online of good quality mRNA profiles on the bioanalyzer and visually compare yours to these. Any size-shift towards the left (smaller average fragment size) indicates some degree of mRNA degradation. Degraded material will also show up as a peak between ~25 and 250nt, so if some samples have much more signal in this region, they are likely more degraded. You should also ALWAYS take an aliquot of your sample before mRNA enrichment and run this on the BioAnalyzer as well, as knowing the integrity of your starting material is very important (and it needs to be consistent between samples, or at least similarly distributed and similarly variant between experimental groups that will be compared.)
There are also some differences between ribodepleted and poly-A selected mRNA profiles on the BioAnalyzer. Assuming good quality starting RNA (RIN 8-9 or higher), poly-A selected mRNA should have strong signal within the mRNA size range (approx 1000-4000nt) but very little signal in the small RNA region (~25-250). Very few transcripts this short are polyadenylated, so if you're seeing signal here, it means you've captured 3' fragments of larger mRNA species which are degraded. Conversely, ribodepletion selectively removes ribosomal RNA while leaving native small RNA in the sample. If the sample quality is high, then you will see a similar profile in the small RNA region of ribodepleted samples as you saw in the input material. Many distinct peaks in the small RNA region is indicative of native small RNA (high sample quality); a single big peak is degraded material. This assumes your samples were cleaned up on a column: any bead cleanup will remove the native small RNA fraction from the sample, as well as most of the degraded material. If a bead cleanup has been performed, you will need to estimate the RNA quality by looking at the intensity and size distribution of the stuff in the mRNA size range, and disregard the small and degraded RNA region, which will be missing.
It is definitely also worth checking RNA integrity by qPCR if your experiment is important and if you can spare an aliquot of sample to do this.
Sorry this is long - advanced bioanalyzer interpretation is complicated - happy to clarify if you have any more questions :)