Deep down, the answer to your question is quite simple, in order to get interference, the light you put in needs to have a chance of passing through both slits. So before you even worry about interference, imagine what the spot created by the light source would like over the screen where you put in the slits, as shown in this diagram. If the beam only grazes one slit, you will simply get diffraction but no double slit interference. If the beam mostly covers one slit but starts grazing the other slit as well, then you will get some double-slit like interference. If the beam is equally distributed over both slits, you will get the maximum degree of two slit interference.

To figure out the answer in more detail, you have to consider the specific light source and geometry you use. For example, if you use an extended source like a light bulb, you will usually need to put a pinhole before the beam as shown here. The effect of the pinhole will be to make the light spatially coherent (meaning that the phase varies in a consistent way across space) and it largely allows you to treat the beam as a uniform plane wave, which varies little in intensity over two closely spaced slits. If on the other hand you use a laser, things are a bit more complicated. Then you would usually treat the light as a Gaussian beam and you would then calculate what the final beam profile would look like at your diffracting screen. But qualitatively everything remains more or less the same - the more equal the distribution of light over the two slits, the stronger the inference you observe will be.