Pulldown resistor value for an NPN transistor to switch a solid state relay
You do need a resistor in series with the base but a better design would also put a resistor in series with the SSR's LED to limit the current unambiguously and not be reliant on hFE of the transistor.

So, I would make the base series resistor 1 kohm and this would force about 2.5 mA into the base. This will likely saturate the collector down to about 0.1 volts so that it is acting like a switch. 

Whatever volt drop is specified by the SSR data sheet for its LED is then subtracted from 5V to determine the voltage across the LED current limiting resistor. The SSR data sheet will inform you of the nominal LED current so then you can calculate this resistor using ohms law.