Is there a correlation between how many hot springs an area has, and how frequent or severe the area's earthquakes are?
Hot springs usually exist in volcanic regions or in areas where there are extensive (normal) faults. The water circulates through the fault zones (basically damaged zones with high permeability) bringing heat from great depths. However these faults don't have to be active.
Therefore a correlation between location of hot springs and present day seismic activity is not necessary but the faults were indeed active at some point in history.
Edit: @winwaed: I wanted to add that there indeed are faults associated with Hot Springs in AR. I am including a figure of the region borrowed from a NPS publication here. Look carefully at the figure in the lower right where water is flowing upwards through a fault. A lot of people have the misconception that there are no faults in the interior of plates. That is not always true. In fact quite the opposite.
Yes, there is a correlation.
In other words: places with hot springs are more likely to experience earthquakes than places without hot springs. The relationship is scale-dependent in time and space (your house might not sit on a hot spring but still get earthquakes).
Ideally I'd dig around and find actual data to quantify this spatial correlation (and I hope someone does!), but for now I will settle for qualitative evidence at the continental scale. I offer a visual correlation between the two datasets in question. First, geothermal springs:
Like any correlation in geoscience, it's not perfect. But, broadly speaking, there is one.
Footnote: Actually, there are at least two correlations. As with fracking for shale gas, well stimulations and other industrial activity in geothermal fields has recently been blamed for induced seismicity in places like the Imperial Valley of southern California. Sometimes correlation does imply causation!
As a counter to @stali's answer, it is possible to have hot springs without volcanism or normal faulting - although these two scenarios do cover the most well known examples.
Hot Springs, Arkansas exists because of the local geological structures (folding rather than faulting) that bring ground water up to the surface from deep in the Earth relatively quickly.
Also radiogenic rocks (eg. granites) can increase the thermal gradient, making this scenario more likely. Eg. the UK's "hot springs". Okay they're not that hot, but shallow boreholes found 60C water at Buxton (and nicely radioactive); and hot water in a Lafarge mine/quarry in Weardale (?). The last I heard, there were plans to turn the latter into a spa...