I tried adding reCaptcha to my signup page in order to prevent automated signups (we're having a problem where someone is creating hundreds of accounts a day to spam).
Adding the captcha resulted in a significant drop in real signups!
Is there a better alternative?
Before I offer some alternatives to captchas, you might want to look into Solve Media. Unlike reCAPTCHA, which is annoying (as most people have to try more than once to get it right), Solve Media doesn't have the legibility problem, and you generate revenue with it.
Some alternatives to not using captcha at all to solve your spam problem:
1. Math problem
Ask to solve a simple math problem like "What is 1+3?".
2. Ask simple, common knowledge questions
E.g. "What color is the sky?"
3. Flag signups from the same IP address
If more than 1 signup come from the same IP address, flag those accounts for manual review.
4. Hell-ban spam accounts
If you delete spam accounts, usually the spammer will just try to create several more. Hell-banning is duping the spammer to think they've won by only displaying their spam content to themselves. i.e. No other visitor can see the spam content, except the spammer.
5. Flag accounts from "temporary email" domains
Here's a massive list of domain names of "temporary email" providers. You should either flag or block signups from these domains. A better option would be to hell-ban signups from these domains by default, otherwise spammers would just try domain names that aren't blocked.
6. Double opt-in
Require new signups to confirm their email address by clicking a link in a confirmation email.
Using a mix of these strategies should be enough to solve your spam signups problem without resorting to using captchas (which degrade the experience for your real customers).
Between yourself and Nishank, you've almost discussed an important point of consideration: Why are you using captchas at all?
You want to avoid "automated signups" because "we're having a problem where someone is creating hundreds of accounts a day to spam".
So the cause of the problem is spammers, and your solution (captchas, of any kind) is to make it difficult for real humans to sign up. The impact of this is significant loss of conversion.
Spam users are your problem, not the problem of your future users, therefore you should figure out how to fix the problem on your end, without impacting genuine users.
Double opt-in and hell-banning are really the only solutions here, with the caveat that double opt-in can have an impact on conversion, but, in some countries there are laws which make this mandatory.