How do you determine how much equity to give in the early days of a startup to investors?
This is a fantastic and difficult question to answer. It's a tricky topic because it has to be judged on an individual basis and agreed to by the team (or at least the majority).
Some people may say, take any offer you can get it is better than continuing to bootstrap. I would argue, that may not be the case. When you are bootstrapping you must be more aware of how each dollar is spent. When you get an investment, you may not be as thoughtful or creative on how you stretch the budget.
Also, don't just see it as the amount of money they are giving you (in terms of the trade off for equity in your venture). Are they also bringing their knowledge of the business world and making it accessible to you? Are they giving you access and recommendations to their network? The team needs to be comfortable with the percentage of equity for the monetary infusion.
Example, if an investor offers $100,000 for 45% of the company's equity some founders may see that as a quick way to grow the business while other founders may balk at giving up such a large percentage of their venture. It is a combination of your team's comfort level, your overall gut reaction and your ability to trust that the investor is trying to better your venture.
It really comes down to what the team agrees is a fair or reasonable trade off between the monetary infusion, networking and resources verses losing some control of the overall business. Once again, great question.