You can run your DNA sample on agarose gel to see, whether you have significant degradation.
If you are interested in contamination, you can make a standard photometric analysis to assess the 260/280 and 260/230 ratios and absorbance at 320 nm on NanoDrop or even something similar to Eppendorf's BioPhotometer.
In case RNA may be an obstacle for some down-stream procedures, with Qubit you can also measure separately DNA and RNA concentrations and then decide if you have too much RNA and it's worth treating your sample with RNase.