It is easiest to cross Doctor Who with fiction that is already within the science fiction genre. To be successful such combinations must fit within the established timeline of the Doctor Who universe, unless you have the characters travelling to alternative realities.
With this in mind here are some ideas for crossovers.
Star Wars/Lexx/Battlestar Galatica
Star Wars is easy to fit within the Doctor Who universe if we place it ‘a long time ago in a galaxy far, far away.’ The simplistic theme of good versus evil fits with early Doctor Who and is distant enough for it not to affect the rest of Doctor Who continuity.
We can assume that by the time that most of the major races rise to power the conflict between the Rebels and Empire is a distant memory. In all likelihood most of their technology and relics will have turned to dust.
Luckily, with their TARDIS, player characters can still travel back and visit the era. They could become caught up in the conflict, fly X-wings and even find they have the potential to be Jedis.
The concept of the Force is not too different from certain species capacity for psychic ability. Time Lords might even have a special talent for it, allowing the Doctor or Time Lord PCs to stand up to the Emperor or Darth Vader.
Taking a different an anti-authoritarian character like the Doctor might have issues with the Jedi Council during the era of the prequels. They are, after all, an organisation that take children away from their parents and find little wrong in mind controlling others or creating clones to serve them. Such a character might engineer their downfall, little suspecting what is to come.
Lexx is another show that takes place in our supposed past. Introducing there are two universes, Light and Dark, the Light universe is conquered by an insect race that rules for generations.
The show depicts a depraved, bleak series of planets and the titular Lexx is a ship capable of blowing up planets. The Dark Universe is worse and that is where Earth is found. This is a series with a very pessimistic view of human nature.
The tone does fit some of the darker Doctor Who novels and the idea that the main characters reside within a living ship is not too dissimilar to the TARDIS. With so much injustice there is plenty that the Doctor or similar characters can try to fix and so much more to struggle against.
Without spoiling too much of the re-imagined Battlestar Galatica, it is eventually established that the series takes place in the distant past. This allows the existence of the 12 colonies without contradicting anything we know about humanities future.
The central conflict between the colonists and their creations the Cylons fits many of the themes of Doctor Who, in particular the idea of man vs machine (the cybermen) and creator vs creation (Daleks and any story with killers robots).
It also reinforces the idea that humanity is resilient, even in the face of its own extinction. The plight of the colonists is surely one that would inspire the Doctor. He might even lend them a hand.
PCs in such a setting may find the paranoia of the fleet, especially when the existence of skin job cylons is made, places them in danger. People might question where these strangers came from. Alternatively, with so many people on board, they might manage to go undetected.
Highlander/Pirates of the Caribbean
The Highlander franchise offers a means in which characters can live through generations. While the Doctor flitters between the years these race of immortals take the slow path. This allow meetings to occur across the decades and centuries, not always in the right order.
This race of immortals could be an off-shoot of mankind or, as ‘Highlander 2’ suggests, a race of aliens. They could be exiled Time Lords, their memories removed and their regeneration process altered so their bodies heal but their appearance remains the same (perhaps a failed attempt to gain infinite regeneration cycles).
A perfect way crossover would be for the Doctor to encounter Jamie in the 21st century, barely aged. It could be revealed that shortly after being returned to his own time period he met his end on the battlefield, only to find that he couldn’t die.
Having an Immortal companion would be little different mechanically from Captain Jack and his own brand of immortality.
Speaking of Captain Jack, the Pirates of the Caribbean provide a historical setting with a slice of fantasy. Although this is attributed to magic and the supernatural it wouldn’t require much to give a science fiction explanation for these occurrences.
This would be perfect for those who enjoyed ‘The Curse of the Black Spot’. The Doctor is a roguish character that can easily follow in the footsteps of Captain Jack, although without being motivated for greed..
Day of the Triffids/Village of the Damned/Quatermas
These very British science fiction films have a lot in common with Doctor Who. Most feature ordinary people coping with something extraordinary. There is usually an expert or scientist who investigates and there is always the threat of death.
‘Remembrance of the Daleks’ has already made the connection to Quatermas and it isn’t a stretch to imagine that other science fictions from the same era could have occurred as well.
In stories where there is a major change to the world, such as with Day of the Triffids, events must be altered so that they either happened on a much smaller scale or that the world eventually recovered and pretended nothing was wrong.
This isn’t too hard to believe considering that the UK was able to convince itself that the events of ‘The Christmas Invasion’ and ‘Stolen Earth/Journey’s End’ didn’t happen.
You might also choose to reveal that any large scale changes were undone by the cracks in time as revealed in ‘The Big Bang’. The player characters are simply visiting a section of time before it has been erased.