In a number of elevator controls I have seen, the coil is rated for the line voltage, or if the line is 208V 3Ph 4W they use Line and Neutral on a 120V coil. If it's a weak system and because the elevator is often the largest motor in the building, sometimes the act of closing the contactor and engaging the motor is the exact thing that causes the voltage drop, that then causes the coil to chatter and that can indeed blow the fuses. The thing is, if this only happens when the line voltage is already low for some other reason, it's pretty difficult to find the problem unless you just happen to be there when it happens.
Start by checking that the coil voltage does indeed match the control circuit voltage as the others mentioned. Then learn everything you can about the control circuit design, i.e. what I said above. If it is a line voltage coil, and maybe even if not, see if anyone can give you a history of when the fuses blow and look for a pattern. For example on one that I worked on, the VD only happened when the maintenance shop was running their air compressor AND someone used the elevator. Thing was, they only used the air compressor a couple times per week and only after hours, so the combination of events was pretty rare. I just happened to be there when they ran the compressor and I saw the lights flicker. When I went over and hit the elevator button, the contactor chattered.
