Ignoring his family's worries, he went and climbed the dangerous mountain alone.

"〜をよそに" expresses acting while ignoring the worries, opposition, or expectations of those around you. Here, the person is ignoring the "family's worries," which are the feelings of others, making "をよそに" the most appropriate choice. "〜もかまわず" describes a state of not caring about things one would normally mind (like getting wet or dirty), but "〜をよそに" has a stronger nuance of specifically ignoring the feelings or opinions of others.

「〜をよそに」は、周囲の心配や反対、期待などを無視して行動することを表します。ここでは「家族の心配」という周りの気持ちを無視しているので、「をよそに」が最も適切です。「〜もかまわず」は、普通なら気にするようなこと（濡れる、汚れる、人の目など）を気にしない様子を表しますが、「〜をよそに」の方が、他者の感情や意見を無視するというニュアンスが強いです。