I remember, at age 14, the first time I heard a piece of "new music"--Robert Dick's Lookout for solo flute, played for me in the context of a lesson, when I could ask questions, by my forward-thinking flute teacher--and my first thought was, "WHOA! THIS IS SO COOL!" It spoke to me on a fundamental level. Ever since this overwhelmingly positive initial experience, I have stayed open to new music, and it has always intrigued me, whether or not I liked it. This makes it easy for me to buy into works that many people think are "strange," "ugly," "crazy," or just plain "bad." I want to understand Jennifer Higdon, Toru Takemitsu, and Jonathan Harvey the same way virtually all musicians want to understand Mozart, Beethoven, and Brahms. (Note: I want to understand those guys too!)
I have heard many times that hatred and fear frequently are born from a lack of understanding, and I think there is some truth to this. It certainly applies to people's attitudes toward contemporary music; I think that new works often turn people off or intimidate them because most folks have never had a chance to engage a new work and understand it. The fact is, understanding new works takes, well, work. Most people, whether musicians or not, have a baseline understanding of western music predating 1900; it might not be their cup of tea, but it just sounds like it makes sense. After all, it is all structured using the same system of tonality as "Twinkle, Twinkle Little Star," "Mary Had a Little Lamb," and virtually every other children's song we have been singing for generations.
Not so with much new music. In addition to atonalism (in all of its guises, from free, to serialism, to aleatoricism, etc.), new music is often replete with "extended techniques," which, honestly, are becoming less and less extended and more and more expected all the time. I understand that if someone has never had the opportunity (or, to be blunt, taken the opportunity for themselves) to try and understand such music, they may well dislike it or feel intimidated by it. In this context, having an open mind means setting aside one's initial reactions to do the work it takes understand a piece. There will be plenty of time later to decide whether it's good or bad, likable or unlikable. In the meantime, in my view, there is a single guiding principle to help the new, new musician navigate a contemporary piece: treat it like it's Mozart.
Would you play Mozart out of tune? With poor finger technique? With a sound not appropriate to classical-era sensibilities? Without dynamics or clear phrasing and articulation? Most likely, the answer is no. Treat any piece of contemporary music the same way: commit first to understanding what is on the page and then to realizing it as best you can. This might mean that you have to dig up some explanatory texts on extended techniques and ask what feel like basic questions ("How do I do this?") of colleagues and teachers. It may mean you have to listen to a bunch of different recordings to hear how other performers have pulled off some gestures that are unfamiliar to you. You will absolutely have to commit to sounding horrible (many times) until you figure out how to execute the score, but you probably sounded horrible on Mozart in the beginning too! (I know I did!) This is commitment: hanging in there through all the slogging until you not only have it, but you can do it over and over again.
Remember that the only thing separating the piece in front of you from Mozart is the language, and with time, patience, and intelligent work (shout-out to Trevor Wye), you can learn that language. This is commitment. Also remember that, like Mozart, the composer of any contemporary work wrote what s/he wrote in the service of expression. This means that there is a reason for everything on the page, from the most bizarre, far-out sounds to commonplace rhythms, meters, pitches, and articulations. Both the new aspects of the musical language and your trusty, old musical fundamentals must be secure in order to render a compelling performance. If you commit to embracing the totality of any piece--new or old--in this way, you will find fear and apprehension falling away. Does this mean you'll love every piece of new music that comes your way? No, but I'd bet the farm you'll love a lot of it, and you'll likely fall in love with the process, resulting in whole, new repertoires to explore.