What does it mean when your destination airport is currently "below IFR minimums"?
It would be very helpful to know where you read or heard the phrase, to get some context. But, it likely means that weather conditions at the airport are below the IFR takeoff minimums in 14 CFR 91.175:


  (f) Civil airport takeoff minimums. This paragraph applies to persons
  operating an aircraft under part 121, 125, 129, or 135 of this
  chapter.
  
  (1) Unless otherwise authorized by the FAA, no pilot may takeoff from
  a civil airport under IFR unless the weather conditions at time of
  takeoff are at or above the weather minimums for IFR takeoff
  prescribed for that airport under part 97 of this chapter.
  
  (2) If takeoff weather minimums are not prescribed under part 97 of
  this chapter for a particular airport, the following weather minimums
  apply to takeoffs under IFR:
  
  (i) For aircraft, other than helicopters, having two engines or less—1
  statute mile visibility.
  
  (ii) For aircraft having more than two engines— 1⁄2 statute mile
  visibility.
  
  (iii) For helicopters— 1⁄2 statute mile visibility.


Landing minimums are per-approach so it wouldn't make sense to say that the whole airport is below IFR minimums based on landing criteria. However, without reading or hearing the phrase in context, I may be wrong.

As for landing there anyway, even in zero visibility a private pilot operating under part 91 is free to take off, or to start an instrument approach. You can't legally land if visibility is below the approach requirements, but you can still fly the approach to 'take a look'. It might not always be smart, but it's legal.

On the other hand, commercial flights under part 135 or 121 can't even start an approach unless conditions are above certain minimums (e.g. see 135.225(a)). Those minimums are usually just the ones required for the particular approach, but they can also have operator-specific requirements. Of course, many commercial aircraft have crew and technical capabilities that light aircraft don't (like shooting a CAT III ILS), so they might still have a better chance of landing in marginal conditions.