How to tell how sensitive a photodiode is?
There's a curve in your device's datasheet that tells you exactly that:
Combined with the line in the Characteristics table that tells you the sensitivity is 0.55 A/W at 850 nm, this gives you the sensitivity at whatever wavelength you are using.
(Note: in the optical communications field, we'd call this a responsivity curve, not a sensitivity curve. For us, sensitivity is a measure of the minimum detectable signal)
if anyone has any good ideas for near-infrared (800-850 nm) photo diodes which are extremely sensitive so have a large range from not much light which is what I need.
Standard p-i-n photodiodes can't produce more than one carrier pair per photon absorbed, so you won't find any with a dramatically stronger response than this one.
If you are working at one specific wavelength, you might be able to find or custom-order one with an anti-reflective (AR) coating for that wavelength and get near a 25% improvement in responsivity (based on the 80% quantum efficiency spec for this device). This will increase the cost.
Some other solutions:
As others have said, you can use an avalanche photodiode (APD), which has an internal current gain process to effectively increase the responsivity above 1 carrier pair per photon. However these require very carefully controlled high-voltage bias, which will increase your system cost.
Or, if your problem is actually focusing your light beam onto the small device area, you could use a larger device (which will be slower and will tend to be more expensive).
Finally, a solution that hasn't been suggested yet in another answer: You could be sure your light source is chosen to match the peak responsivity wavelength of your O/E.
Another comment: The Mouser page you linked calls this a phototransistor, but the Osram datasheet just calls it a photodiode, and its responsivity is typical for photodiodes, not phototransistors. Another way you can get internal gain in the device is to use an actual phototransistor. I don't use these devices regularly, but I expect they might be relatively slow compared to ordinary photodiodes.