Ralph was baptised on 17 April 1806 and the baptism record reads:
1806: Ralph son of John Brown, joiner, Hoydon Dock, parish of Wallsend, Ct N. by his wife Martha was born the 13th March and baptised the 17 April.
Witnesses: present John Bolam and Elisabeth Shotten both of Hoyden Dock.
William's baptism record was more helpful:
1803: William, son of John Brown, joiner, Hoyden Dock by his wife Marrion was born on the 18th August and baptised on the 12th September.
The father was born in the Parish of Alnwick and the mother in the parish of Wooler, her maiden name was Robison.
Witnesses present when the child was baptised John Bolam, ship carpenter and Robt Chaddock both of Hoyden Dock.
Although the mother's name is different, I'm confident that this is the same couple - there's no other John Brown in the Bankside Registers, and there is a marriage between John Brown and Martha Robinson in Alnwick (Northumberland) in 1799.
So far I've not found a baptism for either John or Martha/Marrion - as you might imagine there are quite a few John Browns and I haven't sorted through them yet.
Besides William and Ralph I've found no baptism records for children, but John is on the 1841 census with two young women living with him:
Mary aged 25
Martha aged 15
In the 1841 census no information is given about the relationships between people in a household, and ages are rounded down to the nearest 5 years. The older woman could be a child of the couple, it seems unlikely that the younger one is - Martha would have been between 48 and 53 when the younger Martha was born.
Ralph married in Houghton-le-Spring, Co Durham, England in 1830 and this clue enabled me to find his father on the 1841 census in Easington Lane, part of the old parish of Houghton-le-Spring. I was also able to find their burials at Hetton-le-Hole Church (Hetton parish was carved out of Houghton in 1838, and Easington Lane was in the new Hetton parish). Martha died in October 1839 at the age of 66. John died in May 1850 and his burial record states he was 75 years old.