I thought the "crystal" is an oscilator. It takes in DC and outputs AC as I understood. Correct me if I'm wrong, I'm here to learn.
I think you are confusing crystal oscillators with crystals.
Crystal oscillators often come in these metal cans but usually have 3 or 4 pins - power, ground and the signal output (the 4th pin is unused). Those will produce some sort of AC signal when you power them up.
Crystals have only two pins and there is nothing else inside the can but a piece of quartz in a special holder. This will not do anything by itself, they only act as a very narrow band (and stable) filter/resonator around the frequency printed on the can. You can build an oscillator out of it but it needs some sort of active device to keep the crystal vibrating (think of hitting a gong - unless you keep hitting it it will stop sounding after a while) - typically a transistor oscillator and a something like a buffer stage. That is what you would typically find inside those crystal oscillator cans above.
You also don't want to build a transmitter directly out of a crystal oscillator - those things are designed to drive digital logic, they often produce squarish wave signals. If you feed that to any sort of antenna you will produce a wideband interference/EMI that will cause problems. You could even get a hefty fine for it. The circuit needs a low pass filter (and probably a matching driving stage) at the output in order to eliminate the harmonics inherent in the distorted/square wave signal.
And finally - a 32kHz oscillator would make a terrible transmitter. That's an almost audio frequency, in order to get any kind of reasonable energy transfer the antenna would have to be enormous - the wavelength is around 9km, so a dipole would be 4.5km long. Or you would need huge power levels to compensate for the inevitable losses.
Why do you even want to build a 32kHz transmitter in the first place?