I think you can find your answer giving a look to this google developers page https://developers.google.com/recaptcha/old/docs/asp (Updated Link)
this suggest how implement server side recaptha management.
Here an implementation suggestion
On the submit.asp file:
1) Add on the top the code from the google's developers' page for the challenge writer:
recaptcha_public_key = "your_public_key" ' your public key
Function recaptcha_challenge_writer()
...
End Function
2) add before the input button the following table row:
<tr>
<td valign="middle">
<%=recaptcha_challenge_writer()%>
</td>
</tr>
3) Whatever you want add code to verify and display error message, like:
<%
errorMessage = Request("error")
If (errorMessage <> "" Then
%>
<p class="error-message"><%=error%></p>
<% End If %>
...
<td style="<% If errorMessage<>"" Then Response.Write "color:red" %>">
...
In your sendmail.asp script:
1) Add on the top the code from the google's developers' page for the confirm function:
recaptcha_private_key = "your_private_key" ' your private key
Function recaptcha_confirm(rechallenge,reresponse)
...
End Function
2) then add the code to check the input:
<%
user_lesseename = Request("user_lesseename")
recaptcha_challenge_field = Request("recaptcha_challenge_field")
recaptcha_response_field = Request("recaptcha_response_field")
If (user_lesseename = "") Then
Response.Redirect "caller.asp?error=" & Server.UrlEncode("Lesee name could not be empty")
Else
server_response = recaptcha_confirm(recaptcha_challenge_field, recaptcha_response_field)
If (server_response <> "") Then
Response.Redirect "caller.asp?error=" & Server.UrlEncode("Recaptcha value is not correct: " & server_response)
End If
End If
'The input is validate continue with sendmail code
%>
If you instead prefere to manage recaptcha checking client side you should use AJAX as illustrated in https://developers.google.com/recaptcha/docs/display#AJAX