<apex:page standardController="Task" tabStyle="Activities_VF__tab" extensions="extTaskEdit" showHeader="true">
<apex:form >
<apex:sectionHeader title="My Task" subtitle="{!task.Subject}"></apex:sectionHeader>
<apex:pageBlock title="My Task Edit" id="thePageBlock" mode="edit">
<apex:pageMessages ></apex:pageMessages>
<apex:pageBlockButtons location="both">
<apex:commandButton value="Save" action="{!save}"/>
<apex:commandButton value="Save & New Task" action="{!savenewtask}"/>
<apex:commandButton value="Save & New Event" action="{!savenewevent}"/>
<apex:commandButton value="Check Spelling" onclick="doSpell('en', document.getElementById('{!$Component.descr_section.task__description}'), document.location.protocol + '//' + document.location.host + '/servlet/com.spellchecker.net.sproxy.SProxy', true, undefined, undefined, 'Our SpellChecker feature does not support Internet Explorer 3.0. However,it does support the following browsers:IE 4+,Netscape 4+', 'We are sorry,currently we do not support Netscape Navigator 3.0,Internet Explorer 3.0. Please upgrade your browser', 'There is nothing to check', 'Spell Check Navigation', 'Spell Check Content', 'Spell Check Actions', 'Suggest All', 'Hidden Frame - Ignore');return false;" />
<apex:commandButton value="Cancel" action="{!cancel}"/>
</apex:pageBlockButtons>
. . . . . . . . . . . .

. . . . . . . . . .. .

<apex:pageblockSection title="Reminder" id="reminder_section" columns="1">
<apex:inputhidden id="task__remdatetime" value="{!task.ReminderDateTime}" />
<apex:pageBlockSectionItem id="isRemSet">
<apex:outputLabel value="Reminder" for="task__IsReminderSet" />
<apex:pageBlockSectionItem id="remDetails">
<apex:inputfield id="task__IsReminderSet" value="{!task.IsReminderSet}" onclick="togglereminder(this.checked);" /><!-- -->
<apex:pageBlockSectionItem id="remdatetime">
<apex:inputfield id="task__ReminderDate" value="{!proxyReminderDate.Valid_Until__c}" styleclass="FormDataASmaller" />
<apex:selectList id="task__ReminderTime" value="{!reminderTime}" size="1" styleclass="FormDataASmaller">
<apex:selectOptions value="{!ReminderTimesList}"></apex:selectOptions>
</apex:selectList>
</apex:pageBlockSectionItem>
</apex:pageBlockSectionItem>
</apex:pageBlockSectionItem>
</apex:pageBlockSection>
</apex:pageBlock>
</apex:form>
</apex:page>

Above is the Part of VF code. See here I'm making some adjustment using inputfield, as well as trying to make up for distributed controls for reminders dropdowns.

Below is my Apex Code:

private void doAddReminder()
{
if(taskrecord.IsReminderSet)
{
String stringDate = FormatDate(reminderDate.Valid_Until__c) + ' ' +reminderTime.substring(0,reminderTime.length()-3)+ ':00';
taskrecord.ReminderDateTime = dateTime.valueOf(stringDate);
if(reminderTime.substring(reminderTime.length()-2,reminderTime.length()) == 'PM')
taskrecord.ReminderDateTime = taskrecord.ReminderDateTime.addHours(12);
}
}

private String FormatDate(Date dt) {
string year = String.valueOf(dt.year());
string month = String.valueOf(dt.month());
string day = String.valueOf(dt.day());
string stringDate = year + '-' + month + '-' + day;
return stringDate;
}

Please note that I've now modified the function FormatDate(), after you pointed out that there might be some issues with Date to String conversion. EARLIER it was this INSTEAD of FormatDate function :

String.valueOf(reminderDate.Valid_Until__c)

In above code, please take the assumptions:

reminderTime is ALWAYS in this Format: 12:00 AM to 11:30 PM
reminderDate.Valid_Until__c is a Date value.

I hope it will help, in any case let me know if u need anything else.

Thanks.