(covers information from several alternate timelines)
T'Pol and Archer sharing a blanket
A blanket was a piece of fabric used to cover an individual while sleeping. It could also be used to provide warmth.
While being held hostage by a commando of the Andorian Imperial Guard in the meditation room of a Vulcan monastery at P'Jem in June 2151, Captain Jonathan Archer offered Sub-commander T'Pol to share his blanket with her during night. She first rejected his offer and told him that the cold was preferable to the smell of a Human and it was over twenty-four hours ago since she used her nasal numbing agent. However, she later accepted and shared the blanket with Archer. (ENT: "The Andorian Incident")
The props section on the call sheet listed this blanket as "tattered handwoven blanket".
A blanket used for coverage at the catwalk
The crew of Enterprise NX-01 and the Takret Rellus Tagrim, Renth, and Guri used blankets to separate the compartments at the catwalk for privacy. (ENT: "The Catwalk")
In 2233 of the alternate reality, baby James T. Kirk was wrapped in a receiving blanket before being handed to his mother. (Star Trek)
In 2364, an injured Beverly Crusher asked Jean-Luc Picard if he'd brought a blanket with him to Minos. (TNG: "The Arsenal of Freedom")
In 2367, Susanna Leijten compared her feeling of mental lethargy to having her brain "wrapped up in a blanket". (TNG: "Identity Crisis")
In 2368, after Worf delivered Keiko O'Brien's child, he wrapped her in a blanket before handing her to Keiko. (TNG: "Disaster")
In 2370, Quark of Deep Space 9 set himself up as the promoter of a racquetball match between Julian Bashir and Miles O'Brien on the pretense that the funds were going to monks acting through the Bajoran War Orphans Fund in order to purchase winter blankets, which they had already made a down payment on. After the two discovered that Quark was attempting to rig the match, they put a stop to his scheme, but warned him that they would still hold him to seeing that the monks would receive the blankets regardless of the outcome. (DS9: "Rivals")
Gentry covered with a blanket
The Doctor covered the dead remains of "David Gentry" with a blanket when he performed a post mortem analysis in sickbay. (VOY: "In the Flesh")
Blankets were part of the emergency supplies aboard certain Starfleet shuttlecraft. In 2373, in an illusory scenario created by an alien, a badly injured Kathryn Janeway, stranded on a hostile planet with Chakotay, told him they would need both blankets and rations. (VOY: "Coda")
That same year, while The Doctor's holographic daughter Belle was dying, her brother Jeffrey brought her a blanket. (VOY: "Real Life")
In 2378, Seven of Nine received a blanket from a Ventu girl living on Ledos. When she got lost looking for the remains of a class 2 shuttle in a forest and got cold, the Ventu girl gave the blanket to her to warm her up. Later on board USS Voyager, Commander Chakotay gave it back to Seven. (VOY: "Natural Law")