Influence of active area of photo diode on sensitivity
First, you should be sure you understand the difference between responsivity and sensitivity.

Responsivity is the ratio of the photodetector current to the light impinging on it. The responsivity is given by

$${\mathcal R}=\frac{\eta e P}{h \nu}$$

where \$e\$ is the fundamental charge constant (the magnitude of the charge on an electron), \$P\$ is the optical power falling on the photodiode, \$h\$ is Planck's constant, and \$\nu\$ is the optical frequency (\$h\nu\$ is the energy per photon). The last term, \$\eta\$, is the quantum efficiency, essentially a catch-all term indicating what fraction of the photons reaching the photodiode actually produce current carriers.

Sensitivity, on the other hand, is the minimum amount of light that must be applied to the photodetector to produce a current distinguishable from noise. It's usually specified as a current equal to the noise or dark current of the photodetector (in which case it's often called noise equivalent power or NEP instead of sensitivity). In a receiver for an optical communications system, the sensitivity is usually specified as the minimum signal power required to achieve a certain bit error rate (BER) such as \$10^{-9}\$ or \$10^{-15}\$. Note that lower sensitivity is preferable to higher sensitivity: It's better to be able to detect a 1 nW signal than a 1 mW signal.

Now, responsivity does not depend on the photodiode area. However it will be more difficult to focus a spot from your source onto a smaller detector. 

As for sensitivity, a larger photodiode will, all else being equal, have greater leakage current, and therefore poorer sensitivity. It will also have higher capacitance and therefore lower bandwidth, in case you're using it in a commnunications application.