To edit the display, go to Admin -> Themes -> Manage Templates -> "Front Page" and find this area:
<IF {SWITCH_BIRTHDAYS} and {NUMBIRTHDAYS}>
<div class="box">
<div class="boxtitle" onclick="wsn_minmax('birthdaybox')">{LANG_MAIN_BIRTHDAYS}</div>
<div class="boxbody" id="birthdaybox">
<div class="lefticons"><img src="{IMAGESURL}/birthday.png"></div>
<!-- BEGIN BIRTHDAYS --><a href="{MEMBERPROFILEURL}" class="{MEMBERGROUPNAMESTYLE}">{MEMBERNAME}</a><IF {MEMBERAGE}> ({MEMBERAGE})</IF>, <!-- END BIRTHDAYS -->
</div>
</div>
</IF>
<div class="box">
<div class="boxtitle" onclick="wsn_minmax('birthdaybox')">{LANG_MAIN_BIRTHDAYS}</div>
<div class="boxbody" id="birthdaybox">
<div class="lefticons"><img src="{IMAGESURL}/birthday.png"></div>
<!-- BEGIN BIRTHDAYS --><a href="{MEMBERPROFILEURL}" class="{MEMBERGROUPNAMESTYLE}">{MEMBERNAME}</a><IF {MEMBERAGE}> ({MEMBERAGE})</IF>, <!-- END BIRTHDAYS -->
</div>
</div>
</IF>
If there are no birthdays today, the box will be hidden. If you'd rather show an empty box, just remove {NUMBIRTHDAYS} from the above. You can also use {NUMBIRTHDAYS} to display how many birthdays there are today.
Since "today" varies by timezone, keep in mind that people in differnet parts of the world may see what are from your perspective yesterday's or tomorrow's lists.