If apex:actionSupport is not working for apex:inputField, use apex:actionRegion.

Sample Code:

<apex:actionRegion>
    <apex:inputField value=”{!opportunity.stageName}” id=”pb”>
        <apex:actionSupport event=”onBlur” rerender=”table” status=”status”/>
    </apex:inputField>
</apex:actionRegion>

Also, note possible values of event include “onblur”, “onchange”, “onclick”, “ondblclick”, “onfocus”, “onkeydown”, “onkeypress”, “onkeyup”, “onmousedown”, “onmousemove”, “onmouseout”, “onmouseover”, “onmouseup”, “onselect”, and so on.