It is important that all questions concerning work equipment, liability and costs are clearly set out before starting homeworking.
Deciding who will cover the cost of providing equipment and any additional costs the homeworker incurs, should be covered in a homeworking policy. This is because you can change the policy without having to get the homeworker's consent, whereas you'd need it if those issues are covered in their employment contract. This gives you greater flexibility in case you need to cut costs.
There is, generally, no legal obligation on you to provide the equipment necessary for homeworking. Whether or not you do can depend on whether the homeworker already has the necessary equipment, such as a PC/laptop and an internet connection.
If they don't, you can provide the equipment to them - and this might be preferable if there are security or legal compliance risks, or if they need specific items to perform their duties.
Note that the law requires you to make reasonable adjustments for a disabled homeworker. This means you may need to provide such workers with suitable equipment (or reimburse their cost of obtaining it).
If they do use their own equipment, you need to ensure that it is properly maintained so that it doesn't cause security vulnerabilities and compromise your own data protection obligations.
The homeworker may incur increased costs because they work from home, such as electricity and heating. But there isn't a legal obligation on you to pay or contribute towards this.
If a homeworker hasn't chosen to work from home voluntarily, they canon their extra costs. This could be paid by you as tax-free allowance or the homeworker can claim it themselves.
You will be usually be responsible for any breaches of any civil or criminal laws by the homeworker, such as breaching data protection regulations or anti-bribery and corruption laws. You should provide them with appropriate support, to help prevent such breaches. This can include providing them with appropriate policies and training.