The logic of your if statement needs to be made clearer:

if (take.value.indexOf('@') == -1) 

You can test this by running your current if statement like so:

> !"hello".indexOf("@") > -1 
true
> !"hello@".indexOf("@") > -1 
true

I would probably point out that you should name your variables a bit better. Maybe instead of 'take' you should call it emailID?

function validateEmail() {
    var emailId = document.getElementById('emailID').value;
    var errorMsg = document.getElementById('alert3second');
    if (emailId.indexOf('@') == -1) {
        errorMsg.style.display = 'block';
    } else {
        errorMsg.style.display = 'none';
    }
}

instead of using onkeypress use onkeyup

The onkeyup attribute fires when the user releases a key (on the keyboard). https://www.w3schools.com/tags/ev_onkeyup.asp

and add style="display:none;" to alert3second div to make the message disappears when there is no value in the input

    
function makeHappen() {
    var take = document.getElementById('emailID');
    var a = document.getElementById("alert3second");
    
    if (take.value.indexOf("@") == -1) {
 
        a.style.display = "block"
    } else {
        a.style.display = "none"
    }
}

 <div class="form-group">
 <input type="email" class="form-control font-weight-bold"
    placeholder="YOUR EMAIL ADDRESS" 
    id="emailID" onclick="formValidator2()"
    name="budget3" onkeyup="makeHappen()" required>
    
    <div id="alert3" class="alert-danger">
      <i class="fa fa-warning pr-2 pt-1"></i>
      Please Your email address is required
    </div>
    
    <div id="alert3second" class="alert-danger" style="display:none;">
      <i class="fa fa-warning pr-2 pt-1"></i>
      Enter a valid email
    </div>
</div>