Marriage
In Classical Greece's patriarchical society, women had few legal rights and no political rights. As legal persons, they were the responsibility of their kyrios (lord) for the duration of their lives. When they were young, their father was their kyrios; if their father died while they were still a minor, the role would be transferred to their closest adult male relative. Marriage was the momentous occasion when a girl became part of another household and her husband became her new kyrios.
You can imagine how this would affect the question you're asking. Every enslaved person was considered the property of their despotês (master). Helots, like enslaved people elsewhere in Greece, were not considered legal persons. The concept of marrying one would introduce various legal impossibilities.
If a Spartiate wished to marry their own helot, it would be impossible because it would double up the status of despotês and kyrios. The helot would have to be freed before he or she could marry their former master.
If a Spartiate man wished to marry another Spartiate's female helot, it would be impossible because the woman could not have both a despotês and a kyrios. The helot would have to be freed before she could marry.
If a Spartiate woman wished to marry another Spartiate's male helot, it would be impossible because a freeborn person could not have an unfree person as her kyrios. Apart from the social outrage that would be caused by the very thought of a free Spartiate woman committing herself to the household of a helot, it would create an absurd situation in which the man who was supposed to be legally responsible for his wife was not himself a legal person. Again, the helot would have to be freed before this could happen.
As you can see, in every case the marriage is categorically impossible unless the helot is freed. This was itself impossible; according to a fragment of Ephoros, it was illegal for individual Spartans to free their own helots. Only the Spartan state was able to release enslaved people from their unfree status.
In any case, in Athens it was actually illegal for a citizen to marry a non-citizen, even if they were free. A male citizen who married a foreigner would be fined, and his wife sold into slavery; a female citizen would have her husband sold into slavery and his property confiscated, leaving her penniless. Similar laws are likely to have existed at Sparta, meaning that it would be a crime to marry a helot even if it had been possible to free them.
In short, Spartan society seems to have had something of a triple lock on the possibility of a citizen marrying a helot. It could never happen.
Relationships
While we have no explicit evidence of Spartiate citizens having relations with their helots, there are good reasons to assume that it happened. For one thing, there was a subclass in Spartan society known as the nothoi (bastards) or mothakes, who were not full citizens and were sometimes even born into slavery, but who were nevertheless "sponsored" to go through the Spartan upbringing and learn the ways of citizens. Late sources tell us that several very prominent Spartan commanders belonged to this group. Gylippos, the saviour of Syracuse against Athens, was allegedly a mothax, as was Lysander, the naval commander who orchestrated Athens' final defeat in the Peloponnesian War. The latter is even said to have been made a citizen in recognition of his achievements, which is otherwise almost unheard of in Sparta. Now, why would Spartiate citizens be involved enough with the lives of unfree children in their own household? One likely answer, suggested by the title of nothoi, is that they were in fact their own children by their female helots.
It is important not to equate the existence of nothoi with the existence of romantic relationships. As I said, enslaved women had no legal personhood and no rights; sexual violence against enslaved people was likely rampant in all of Ancient Greece and Sparta would be no exception. Without specific testimony, we cannot tell if the children sent through the Spartan education were the illegitimate children of Spartiates and their helots; if we assume they were, we cannot tell if they were the product of consensual relationships or sexual exploitation. All Spartans involved in such situations have managed to keep it off the historical record. Either way, it is likely that enslaved people of either sex in Spartan households were indeed the object of Spartiate sexual desires, even if our only evidence is the final result of that fact.
EDIT: got my punishments mixed up