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I have a somewhat complex answer that I hope you find helpful! :) I think you need to look at what other advertising mediums in your market cost and then figure out if you can offer a competitive CPM (cost per thousand impressions).

I would suggest that you try to calculate the number of impressions your partners could expect. Most partners collaborate and contribute to museums and non-profits as a marketing or PR function of their business, so using the language that they are used to will help facilitate an apples-to-apples comparison between your opportunities and others they might be considering. Non-profits almost never do this, so it is hard to make the case for sponsorship dollars within companies. 

You can begin to do this by calculating the number of potential impressions from your website or email newsletter, divide that by 1000. Take the result and divide it by the partner's proposed investment. This will give you a cost per thousand impressions or CPM. This is the standard measure that most marketers will use to evaluate the cost of various marketing and PR channels.

If you can make an educated guess about the number of clicks or purchases a partner can expect, that's even better! But, that can be trickier if you don't have any trend data yet. Starting with CPM is a good first step and then try to collect conversion data to help enrich your reporting and estimating in the future.

Your goal should be to have a CPM that is competitive with other channels in your market. So, for example, if TV has a CPM of $25, your goal would be to have something competitive to (or less than) that.

Another idea is to create some sharable audience profiles or personas that show your partners the target audiences they are reaching. If your audience is hyper-relevant to their business, this can really help convince a partner to spend more money with you to reach a smaller audience because they are more targeted.

Best of luck with your project!