Airports don't typically close because of visibility. In fact, ATC won't stop a pilot from flying an approach just because the weather is below minimums. There are a couple of reasons for this:
The determining factor for landing is flight visibility, which may be different from the ground visibility that ATC has access to. Under US Part 91 rules, a pilot may fly an approach even when the reported weather is below minimums, and if they have the required flight visibility and are in a position to land, they are allowed to land. Part 121/135 operators aren't even allowed to try the approach though. ATC does not have to query the pilots to see which rules apply to them. They assume that they won't ask for/accept a clearance if they aren't allowed to.
Different aircraft have different equipment and different pilots/operators have different training and rules. For instance, one operator may be approved to use a Heads Up Display to land when the visibility is below landing minimums for another operator that does not have one. ATC assumes that the pilot knows what they are doing and will only do what is legal for them.
As far as this specific case, @JayCarr has done a good job of answering what heppened here. The pilots simply flew the approach incorrectly....
In regards to the title of your question (which is a little bit different from the body of your question), each approach has the minimum flight visibility listed for it at the bottom of the approach plate. Unless specific authorization is received to reduce the visibility (as in the HUD example above), no it is not legal to land if the visibility is below minimums.
Each approach procedure also lists minimum altitudes that you may fly to at various points during the approach. If an ILS is out of service, there is very often another type of approach that may be flown that usually prevents the pilot from descending as far.
A very common type of approach plate will have an ILS approach combined with a Localizer approach. Since an ILS is made up of a localizer for horizontal guidance and a glideslope for vertical guidance, if the glideslope becomes inoperative then the localizer can still be used on its own. A localizer only approach without vertical guidance, uses "step-down fixes" to indicate when descents are allowed. Basically, you determine when one of these fixes is crossed and you are now allowed to descend to the next minimum altitude. From the side it looks kind of like a staircase with the last "step" being flown visually in order to land.
An example of the bottom portion of an ILS or LOC approach plate: