If you are talking about viewing the full site on a mobile device, versus the mobile style, this would require you to add some code to detect if it is a mobile device, and then wrap the welcome message with a conditional.
Off the top of my head this is not a easy task, but I have not looked into it.
--------------- Added [DATE]1379808632[/DATE] at [TIME]1379808632[/TIME] ---------------
Actually looking around you can use this mod,
https://vborg.vbsupport.ru/showthread.php?t=292704
You will have to edit the plugin,
Mobile Browser Check found in the group
Mobile Browser Detector (BOP5)
In the bottom of that plugin add the following.
At the bottom add lines such as:
PHP Code:
if ($vbulletin->detect->isMobile())
$vbulletin->options['isMobile'] = 1;
Then in a template you can go in you header template and find this code.
Code:
<li class="welcomelink">{vb:rawphrase welcome_x_link_y, {vb:raw bbuserinfo.username}, {vb:link member, {vb:raw bbuserinfo}}}</li>
And change it to this code.
Code:
<vb:if condition="$vboptions['isMobile']">
<li class="welcomelink">{vb:rawphrase welcome_x_link_y, {vb:raw bbuserinfo.username}, {vb:link member, {vb:raw bbuserinfo}}}</li>
</vb:if>
This is completely untested but in theory it should work.