Sometimes does not start

We have the Key-less Entry Remote, but recently the car doesn't always start. The instrument panel comes on like it's supposed to but the car sometimes won't start. When it doesn't, there is a little red car emblem with a key symbol on it that starts flashing in the upper right of the instrument panel. I assume this to mean it isn't reading the key fob. I have changed the battery in the fob itself to no apparent effect. Starting is hit or miss at this point.