How do you guys politely decline Headhunter opportunities while staying in touch to recruit with them in the future?

Not in AM but also dealing with a fair number of HHs reaching out each week. I too want to stay where I am. You can take 1 of 3 approaches in my personal opinion: 

1. Ignore them all unless you come across a position you really want: I did this for the better part of the past year. I just started a new role and was determined to stick it out for two years so I ignored the influx of emails and LI messages from recruiters altogether. My logic was if I am desirable now I will likely be just as desirable in a year, and can expect a comparable volume of HHs reaching out with new opportunities. However, eventually, I came across a position I really wanted (Corp Dev role within my specialty industry), and I immediately connected and started the interview process. Even after I switched industries I was still receiving the same volume and type of HH requests.

2. Answer each of their cold outreach requests to connect: I have done this once or twice with guys who mention they specifically recruit within niche areas of finance I am interested in. However, I would advise against doing this to all recruiters who reach out to you right now as most are SLAMMED. I am receiving midnight and 5 am emails from HHs regularly, some on saturdays. Yes they arent in IB or PE, but they dont make IB or PE money, and they are being worked to death right now. 

3. Ignore them altogether: To be honest, since I switched jobs and have no intent to leave to go anywhere, this is what I've been doing now. For context most of the HHs are reaching about IB Associate roles with a few PE / Credit fund Associate roles sprinkled in. I know they are reaching out to a ton of people, and that I am not special. Therefore, I am not making any effort to follow up. Now if I were a Manager in Corp Dev and had recruiters who reached out specifying that they focused on my specific industry or solely head hunted for Corp Dev Directors, then I would start the conversation regardless.

After writing this out, 1 and 3 seem nearly identical. I guess I would state the difference is in #1 it would be more of an approach for if you could be pulled away currently if a HH reached out with a God-tier position you would take immediately, whereas #3 is more applicable if you are interested in establishing relationships with key HHs that specialize in your specific finance niche. The result is of course the same, start the conversation as it could lead to a new exciting opportunity, either immediately or somewhere down the road.   

In terms of what to actually say to them: Be honest and state that you appreciate them reaching out, and you are looking to establish a relationship. Give them your timeline for when you are open to pursuing new roles, and the types of roles you are targeting. They may ask for background info (GPA, return offers for SAs, etc.), I see no reason not to provide them with this information. Also, be prepared to discuss compensation. Remember, these guys want to add you to their list of folks to reach out to in the future. They want to create an open dialogue with you in the same way an Investment bank does with a PE fund. They want to know that they can rely on you to pick up the phone and consider a position that meets your criteria when they find it down the road.