It's very important in any job hire that you consider the full compensation package, and not merely the rate. It could be that they are asking you to take an unreasonable hit in pay to pad the recruiters pockets and reputation - or it could be that you are getting an absolutely fantastic deal. A few things are important to consider.
Change in Benefits - insurance, 401(k) employer contributions, paid time off
In the new deal are you going from "none of these" to "all of these"? A good employer-supported insurance plan by itself (at least in the US) can be worth more than the $4000 you are quoted.
401k employer contributions, of which 3% cap is somewhat common (but not the min or max), could be worth half of the drop in pay rate by itself ($1,800 of as close to you can get of "free money" if you offered a 60k salary)...or the offer could offer little to nothing.
How about paid time off, in the form of vacation or sick days? If you get a week of paid time off that could easily be worth more than $1000 to you by itself.
Change In Working Hours
This doesn't apply to everyone but sometimes contractors work very different hours than regular employees. If you are moving from hourly contracted to salary, also consider any potential impact of overtime pay if you are getting that
Change in Billing Rate
Another concern here is if you were being billed to the company at, say, twice the rate you were being paid, and now upon hire the recruiter wants you to take an offer that is actually less pay for you compared to the rate you were personally being paid before (which was already much less than the recruiter was collecting). If this is the sort of setup you were working under (which is usually more like temp-to-hire, but I've heard it called contract-to-hire as well), I'd be much more concerned about what is being offered, because it sounds like the recruiter might be trying to low-ball you in exchange for the same (or even more) pay for them and increased reputation for delivering full time employees for such a cheap rate.
If the relationship to the recruiter and you is ending then they lose much of their incentive to try to give you a deal you like, because all they need to do is place you to collect their extra pay (often a lump sum on hire, or shortly after). A more typical setup is for the recruiter to get a percentage of the yearly rate, but some recruiters take a flat fee or even negotiate a bonus for themselves based on how low a rate they can provide employees for.
What Can You Do?
Well, that really depends on the relationship between the recruiter and the company. If the contract is with the recruiter and the company only wants to deal with the recruiter (which is often one reason companies hire recruiters in the first place, to avoid dealing with prospective employee negotiations), then the company might simply be unwilling to negotiate with you. Being in a 3-party negotiation is weird.
Ultimately your ability to negotiate comes down to your willingness to say no and walk away. If you aren't willing to walk away and decline the offer, then the most you can safely do is softly push back with the recruiter and express your dislike for the pay cut. I suppose it's possible that they'd do something, but often if they are game players they'll know enough to know that you aren't willing to walk away, but you can try.
If you are willing to walk away then you have a lot of room to negotiate, and can simply refuse the offer and express that you want the job - but are not willing to take an offer that cuts your take-home pay on day 1. You could try to demand a pay increase (harder), or you could simply say you will only accept an offer if it comes with no decrease in pay.
You will have to deal with the fact that the employer/recruiter can simply decline and make the offer take-it-or-leave-it, or rescind the offer. We can't make that decision for you, but I would generally suggest you look over all aspects of the offer before you turn something down that might actually be a great deal for you.
There is usually room to negotiate, but not all the time and not with everyone. You can also usually soft negotiate and attempt to get a better offer with minimal danger, but that is less than 100%, and success rates are also less than 100%. You can always try, but again - you have to be OK with walking away and finding another job. Not everyone is in a position to do that, so you have to decide for yourself just how much risk you are willing to take to try to get a better offer.