If by searchbox you mean:

<input name="search" type="text" value="<TMPL_VAR NAME="SEARCH">" />
[download]

You'll want to add ESCAPE="HTML" to the TMPL_VAR tag in order to convert quotes into it's HTML equivalent (").

<input name="search" type="text" value="<TMPL_VAR NAME="SEARCH" ESCAPE
+="HTML">" />
[download]

HTH

Update: This problem (with input boxes) is mentioned in the docs under TMPL_VAR

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