What sampling method should we use?
You can use both simple random or stratified sampling in this case, but one might be more appropriate than the other in specific circumstances, especially if these circumstances can be anticipated in advance, since once you implement your sampling plan, it's almost impossible to change.
Let's walk through the problem.
A simple random sample will certainly be easy to conduct here given that you have access to a frame. What will stratified sampling add to this?
Stratified sampling will protect against a "bad" sample. Let me explain. Collecting a simple random sample is risky because the randomness might produce a sample that is, in its nature, special even if it is random. For example, a random sample of 100 selected from a population of 1,000 males and 1,000 females is still quite possible to consist mostly of males or mostly females. In your example, a simple random sample might result in a group of respondents from only a handful of departments. Many people might not find this sample representative for their needs.
There might ba a need to establish precise results for specific subgroups of the population, not just the entire population as a whole. For example, you might wish to estimate the outcome in males versus females, or for various race/ethnicity classes, or by department, or by degree within department. In this case, the subgroups constitute the strata.
A stratified sample is less costly and more easily administered than a simple random sample. In your example above, say that departments constitute the strata. It is much easier for you to contact students by department rather than centrally.
Stratified sampling often gives more precise estimates for population means because the variance within each stratum is oftem lower than the variance of the entire population. Samples within a stratum are more alike.
it's useful to have an equal probability to choose each graduate
I'm not so sure about your use of the word "useful" here. In simple random sampling, the probability of inclusion of an observation $i$ is simply $\pi_i = n/N$ and is the same for all observations. In stratified sampling, the probability of inclusion of observation $i$ from stratum $h$ is $\pi_{ih} = n_h/N_h$. In either case the probability is known and weights can be derived. Both cases are equally useful, I think.
our population is small, known and homogeneous
I'd be careful about making an assumption about the homogeneity of the popualtion. I'm not sure that there is any evidence for it. I'm thinking that that's the entire point of the study: to determine whether there is, in fact, differences across the university. Let me put it another way. In cooking, chefs only need to take a small sip from an entire pot of soup to determine whether it's seasoned well. However, they stir the soup pot thoroughly first to ensure homogeneity of mixture. Now, you notice a pot of soup and wish to sample it to test its flavour. You wouldn't assume, off hand, that the pot was already mixed thoroughly. There's a similar situation happening here.
Sampling is a fascinating topic and you're certainly on your way to learning more about it. I wish you the best of luck.
