It depends if they were criminal convictions or not, and the time elapsed between the conviction and visa application, re:
"As long as you declare the convictions and include the necessary paperwork, it shouldn't be an issue. They can only refuse the visa based on convictions if they happened in the last 12 months."
There are detailed guidance notes on the internet about criminal convictions for spouse visas at least. I would think they apply to other settlement visas also.
This is for more recent criminal convictions:
Basically, they say that if there was a non-custodial sentence handed down, if may affect the decision of the ECO's to give the applicant a visa. Generally, it would depend on the circumstances of the case. But they do say that the conviction (eg a fine) could be a ground for a refusal. How long for is something they do not say. However, if you look at the same guidance notes they state that in the case of a prison sentence, if the prison sentence was a small one, eg for 30 days, it will prejudice their giving a settlement visa for up to 2 years (from memory.)
The longer the prison sentence, the more problems the applicant has in convincing the ECOs to give a visa, within set time frames. I think for a long prison sentence, they will not be awarded a settlement visa for 10 years. The sentences and times involved are on the internet, and I've seen three of them from 2015-6, all with the same information.
The upshot is that a criminal conviction will be something they will take note of, especially if it's a quite recent one.
Article 8 can be used, and the ECO guidance notes do say this. Maybe to offset the negative effect of a more recent criminal conviction.