Be thankful you have someone who cares enough to tell you the truth and good for you for not being defensive about it. Crying at the smallest things and all the time is not going to serve you well. The main solution involves not repressing your feelings, but learning how to moderate them.
You can be sensitive all you want. But, breaking down into crying fits is only going to make you vulnerable, pathetic (as your family member rightly advised), and socially shunned. It's exhausting having to calm down someone every other second, so most people will just avoid you.
You need a healthy dose of self-reflection. The next time you start crying, ask yourself why you are so upset. You will be surprised just how much you know as to the reason. You literally need to reason with yourself that crying is not an appropriate response, no more than an 18-21-year-old is having a temper tantrum when things don't go all your way. Over time, intervening in that way, will break the cycle of breaking out in tears.
And, be sure to thank your family member for giving you honest feedback. It could just end up being a lifesaver.