like kpitonak already wrote: hide the button in qa-styles.css:
.qa-vote-down-button,.qa-vote-down-hover,.qa-vote-down-disabled {background:url(vote-buttons.png) no-repeat; border:0; color:#f1c96b; font-size:1px; height:29px; width:27px; visibility:hidden;}
Additionally you may want to hide the gave + got #number of downvotes located in the userprofiles by:
add this to qa-styles.css:
.qa-uf-userprofile-downvotes {visibility:hidden;}
change the qa-page-user.php:
search for $votegavevalue=... and replace this line by:
$votegavevalue.= "<SPAN CLASS='qa-uf-userprofile-downvotes'>, ".(($downvotes==1) ? qa_lang_html_sub('profile/1_down_vote', $innervalue, '1')
: qa_lang_html_sub('profile/x_down_votes', $innervalue)).'</SPAN>';
delete the .', ' in line:
$votegotvalue=((@$userpoints['upvoteds']==1) ? qa_lang_html_sub('profile/1_up_vote', $innervalue, '1') : qa_lang_html_sub('profile/x_up_votes', $innervalue)).', ';
replace the line $votegotvalue=... by:
$votegotvalue.="<SPAN CLASS='qa-uf-userprofile-downvotes'>, ".((@$userpoints['downvoteds']==1) ? qa_lang_html_sub('profile/1_down_vote', $innervalue, '1')
: qa_lang_html_sub('profile/x_down_votes', $innervalue)).'</SPAN>';
a good reason to hide this vote-down button could be to avoid some kind of mobbing - or demotivation by getting downvotes...