s_day = $("select[id ^='form-widgets'][id $='start-day']")

if(s_day.length > 0){
    console.log(s_day[0]);
}

<select class="day date-widget required date-field" name="form.widgets.entry_start-day" id="form-widgets-entry_start-day">

console.log(s_day[0].val())

s_day[0].val is not  a function

s_day = $("#form-widgets-IEventBasic-start-day");
console.log(s_day.val());

s_day = $(select[id ^='form-widgets'][id $='start-day']);
if(s_day.length > 0){
    console.log($("#"+s_day[0].id).val());
}

var s_day = $('select[id^="form-widgets"][id$="start-day"]:first).val();