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Mindfulness will increase with correct, consistent practice. At some point, if continued practice occurs, the sleep cycle will be affected too, in the form of a lighter sleep with less dreaming and more awareness.
When balancing the Five Spiritual Faculties, Mindfulness is the only factor that is not balanced meaning one can cultivate as much mindfulness as possible without creating any imbalances in the mind.
I would recommend watching the video "How Mindfulness Creates Understanding (The Buddhist TV)" by Ven. Yuttadhammo.
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The better you get at mindfulness while you are awake, the less dreams you will see while you are asleep. When you go to sleep, take the lying posture as the primary object of meditation and practice mindfulness until you fall asleep. If you still keep dreaming on a daily basis, you should get advice on your meditation technique and get it readjusted.