At present we follow giving Dowry to the grooms which is considered as social evil. What Lord's do is considered right because they will have a particuar reason behind it. So I want to know what they followed and why.
The exact word mentioned in Valmiki Ramayan is Kanyaa Dhanam (कन्या धनम्). It was wilingfully (not demanded by grooms) given by king Janaka to his daughters.
अथ राजा विदेहानाम् ददौ कन्या धनम् बहु |
गवाम् शत सहस्राणि बहूनि मिथिलेश्वरः || १-७४-३
कंबलानाम् च मुख्यानाम् क्षौमान् कोटि अंबराणि च |
हस्ति अश्व रथ पादातम् दिव्य रूपम् स्वलंकृतम् || १-७४-४
ददौ कन्या शतम् तासाम् दासी दासम् अनुत्तमम् |
Then that king Janaka of Mithila, the one from Videha lineage, gave innumerable kanyaa dhanam (girl's [patrimonial] riches). He has also given umpteen number of cows, millions of excellent shawls and silk dresses, and elephants, horses, chariots, foot soldiers, besides hundreds of highly decorated girls, divine in their mien, as unexcelled chambermaids and handmaidens to the brides. [1-74-3, 4, 5a]
हिरण्यस्य सुवर्णस्य मुक्तानाम् विद्रुमस्य च || १-७४-५
ददौ राजा सुसंहृष्टः कन्या धनम् अनुत्तमम् |
King Janaka gave beau idéal bridal riches in gold, silver, pearls and corals even, for he is very highly gladdened as Seetha's marriage came true
Shloka says that it was given to the brides and not to the grooms. So it was not Varakatnam (as defined in your Question) but something similar to this with slight variations.