First - your "2m dipole" antenna and "quarter wave" antenna are almost the same thing. A "quarter-wave" antenna is really just a "dipole antenna" with the bottom/ground element bent at an angle - usually 90 degrees. This might also be referred to as a "ground plane" antenna. If you took this "bent" antenna, and straightened it out, it would be a "dipole antenna". =)
Second - both a "2m dipole" and "quater wave" (or ground plane) antenna are, indeed, omnidirectional. Your high altitude balloon (HAB) payload should have absolutely no trouble being heard on APRS once it reaches about 300-500 feet above the local terrain.
My suggestion for which antenna to use on your HAB is to use a simple wire dipole antenna. For 2 meter APRS on 144.390 MHz this would be two, 19-1/2 inch pieces of wire (AWG 28 enameled copper wire works fine), with the driven element (the one connected to the radio's RF output/connector center pin) pointing DOWN towards the earth, and the other element (the one connected to the radio's ground) pointing UP.
The element hanging down (driven element) does not need any support - gravity will do the trick. You can use some tape to secure the element pointing UP to the rigging lines that attach the payload to the parachute/balloon.
You want the DRIVEN element pointing DOWN so that when the balloon pops, all the rigging lines between the payload and the parachute/balloon don't tangle up and bend the DRIVEN element. Don't be concerned about the tangling or twisting/breaking of the ground wire for the antenna - the radio signal will still get out!
Pointing the DRIVEN element DOWN from the payload gives your payload a much better chance of continuing to transmit data during the descent - especially if your HAB traverses the thinner portions of the atmosphere (e.g. anything above 40,000 feet), where the drag effects of the parachute are not yet significant.
If you can afford the payload weight, you may even chose to use a 2 meter "rubber duckie" antenna, coupled with a single 19-1/2" AWG 28 ground wire. My colleague and I have launched multiple HAB APRS payloads using this configuration, and have had no trouble being heard on 2m APRS using only 1/4 watt! Again - if you can afford the weight, using a "rubber duckie" may be a good compromise between potentially losing the antenna due to tangling with the rigging (or impact with the ground), and good signal reception at APRS distances.
Using the "rubber duckie" and a wire ground element, my colleague and I were able to directly receive our 1/4 watt 2m APRS signal from a distance of over 125 miles, and at no time during our HAB's 500 mile journey was it ever out of range of the APRS network. The website aprs.fi reported that our payload could be heard out to a distance of 250 miles (a 500 mile diameter ground circle!) when we were transmitting 2m APRS from 60,000+ feet! (Don't forget to change your APRS path to WIDE2-1 while at altitude!)
By the way, we chose to implement two-way commanding over APRS (using APRS directed messages) between the ground and our balloon, and the APRS network faithfully relayed all of our commands/messages to the payload throughout its flight. The payload had no problems receiving our commands over both the wire dipole and the "rubber duckie" antennas. So omnidirectional / signal strength issues were not of concern to us.