The basic obligation is only to recite blessings loud enough that you can hear them, as brought by the Rambam in Mishnah Torah, Hilchos Berachos, Chapter One, the beginning of Paragraph Seven:
כל הברכות כולן צריך שישמיע לאזנו מה שהוא אומר
A person should recite all the blessings loud enough for him to hear what he is saying
However it is often considered worthwhile and praiseworthy to say blessings out loud.
An article on aish.com paraphases Mishnah Berurah 185.3 stating:
it is better to say [a bracha] aloud, as this is extremely helpful in focusing one's concentration.
According to an article on ohr.edu (Ohr Somayach), Rabbi Yitzchak Zilberstein, Rav of the Ramat Elchanan brought in his work Aleinu Leshabeiach the following points:
- If any of the people hearing your blessing will answer Amen you should definitely say it aloud in order to provide them with the opportunity.
- Even if no one will thus respond, it is still worthwhile saying the blessing aloud because doing so serves as a sanctification of the Name of G-d.
- The only time you should refrain from saying a blessing aloud is if there is a hostile listener who will exploit your action to mock religious observance.
As a counter balance to even this third point, the Kitzur Shulchan Aruch 1.3 quotes Avot 5.20 explaining:
"Strong like the leopard" means that one must not be ashamed by people
who mock his service of the Lord, blessed be His Name.
Lastly, the Jewish blogosphere has many examples of encouragement regarding reciting of blessing out loud, as examples:
Obviously you must judge the situation and decide the best course regarding blessing aloud. I hope the information above helps you decide.
In light of Gavriel's comment I went in search of concurring source material. I found in Ben Ish Hai, Halachot 1st Year, Masei 15:
והזוהר הקדוש החמיר מאוד בעונש השומע ברכה ואינו עונה אמן לכן אם המברך על איזה דבר לעצמו רואה דהשומעין מזלזלים באמן ואין עונין, טוב שיברך בלחש כדי שלא ישמעו הברכה ויבואו לידי מכשול
Which can be roughly translated (by me) as:
The holy Zohar indicates very severe punishment for one who hears a blessing and doesn't respond Amen. Therefore, if the blesser sees the listeners are lax in responding Amen, better he should bless quietly so they don't hear the blessing lest it become an obstacle.